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Papers:

Giselle Sosa Jones and C. Trenchea, "Discrete energy balance equation via a symplectic second-order method for two-phase flow in porous media", Applied Mathematics and Computation, Vol. 480 (2024), Paper No. 128909.

Abstract. We propose and analyze a second-order partitioned time-stepping method for a two-phase flow problem in porous media. The algorithm is based on a refactorization of Cauchy's one-leg $θ$ -method. The main part consists of the implicit backward Euler method on $[t^{n}, t^{n + θ}]$ , while part two uses a linear extrapolation on $[t^{n + θ}, t^{n + 1}]$ to obtain the solution at $t^{n + 1}$ , equivalent to the forward Euler method. In the backward Euler step, the decoupled equations are solved iteratively. We prove that the iterations converge linearly to the solution of the coupled problem, under some conditions on the data. When $θ = 1 / 2$ , the algorithm is equivalent to the symplectic midpoint method. In the absence of the chain rule for time-discrete setting, we approximate the change in the free energy by the product of a second-order accurate discrete gradient (chemical potential) and the one-step increment of the state variables. Similar to the continuous case, we also prove a discrete Gibbs free energy balance equation, without numerical dissipation. In the numerical tests we compare this implicit midpoint method with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behaviour and energy approximation, for small and large values of the time step.

Madeline Edwards, Martina Bukač, and C. Trenchea, "A second-order partitioned method for bioconvective flows with concentration dependent viscosity", Annals of Mathematical Sciences and Applications, Vol. 9 No. 1, (2024).

Abstract. This work is focused on the mathematical and computational modeling of bioconvection, which describes the mixing of fluid and micro-organisms exhibiting negative geotaxis movement under the force of gravity. The collective population moves towards the surface of the fluid, generating a Rayleigh–Taylor instability, where initial fingers of organisms plummet to the bottom. The inherent drive to swim vertically generates large collective flow patterns that persist in time. We model the flow using the Navier-Stokes equations for an incompressible, viscous fluid, coupled with the transport equation describing the concentration of the micro-organisms. We use a nonlinear semigroup approach to prove the existence of solutions. We propose a partitioned, second-order, time adaptive numerical method based on the Cauchy’s one-legged ‘θ -like’ scheme. We prove that the method is energy-stable, and for small time steps, the iterative procedure in the partitioned algorithm is linearly convergent. The numerical results confirm the expected second-order of accuracy. We also present a computational study of a chaotic system describing bioconvection of motile flagellates.

William Layton, Wenlong Pei, and C. Trenchea, "Time step adaptivity in the method of Dahlquist, Liniger and Nevanlinna", Advances in Computational Science and Engineering, 1 (2023), no, 3, 320-350.

Abstract. Dahlquist, Liniger and Nevanlinna devised a family of one-leg two-step methods (DLN) that is second order, A- and G- stable for arbitrary, non-uniform time steps. The DLN method thus has strong potential for use in adaptive codes, but the adaptive step size selection is little explored. This report develops two approaches for the efficient local error estimation in the DLN method, and tests their use in a standard adaptivity framework. Many methods of error estimation are possible; herein we focus on ones which involve minimal extra storage and computations. First we evaluate the local truncation error of the DLN method by Milne’s device, using the difference between the DLN and the variable-step second-order Adams-Bashforth (AB2) solutions. Second, we use a recent refactorization of the DLN method, which eases implementation of DLN in legacy codes, to obtain an effective error estimation at no extra cost. We perform a number of numerical tests, comparing the two time adaptive DLN algorithms with some standard numerical ODE packages.

Martina Bukač, Guosheng Fu, Anyastassia Seboldt, and C. Trenchea, "Time-adaptive partitioned method for fluid-structure interaction problems with thick structures", Journal of Computational Physics, 473 (2023), Paper No. 111708.

Abstract. In this paper, we present an adaptive time-stepping numerical scheme for solving a fluid-structure interaction (FSI) problem. The viscous, incompressible fluid is described using the Navier-Stokes equations expressed in an Arbitrary Lagrangian Eulerian (ALE) form while the elastic structure is modeled using elastodynamic equations. We implement a partitioned scheme based on the Robin-Robin coupling conditions at the interface, combined with the refactorization of Cauchy’s one-legged $θ$ -like method with adaptive time-stepping. The method is unconditionally stable, and for $θ$ = $\frac{1}{2}$ , it corresponds to the midpoint rule, which features second-order convergence in time. The focus of this paper is to explore the parameters used in the time-adaptivity, and to compare the adaptive approach to the one where a fixed time step is used. For computing the local truncation error (LTE), which is used to define convergence of the solution, we consider two methods: a modified Adams-Bashforth two-step method and Taylor’s method. The performance of the method is explored in numerical examples. We present an example based on the method of manufactured solutions, where the effect of different parameters is studied, followed by a classical benchmark problem of a flow around a rigid cylinder attached to a nonlinearly elastic bar inside a two-dimensional channel. Finally, we present a three-dimensional, simplified example of blood flow in a compliant artery.

Aziz Takhirov and C. Trenchea, "Efficient nonlinear filter stabilization of the Leray-α model", Journal of Computational Physics, 471 (2022), Paper No. 111668.

Abstract. In this work we study a novel adaptive nonlinear filtering applied to the Leray- $α$ model. Unlike its classical counterpart, the new filtering requires the solution of a linear elliptic problem, with constant coefficients, at each time step. The action of the adaptive nonlinear filter throughout the integration time is refactorized as the solution of a linear system, with the same matrix and multiple right hand-sides. We discuss the theoretical properties of the new filtering approach applied to the BDF2 approximation of the Leray- $α$ model. Numerical tests demonstrate that the filtering damps the high wave number modes of the solution and has similar level of accuracy with the classical filter. Some benchmark results are also presented.

Martina Bukač and C. Trenchea, "Adaptive, second-order, unconditionally stable partitioned method for fluid-structure interaction", Computer Methods in Applied Mechanics and Engineering 393 (2022), 114847.

Abstract. We propose a novel, time adaptive, strongly-coupled partitioned method for the interaction between a viscous, incompressible fluid and a thin elastic structure. The time integration is based on the refactorized Cauchy's one--legged ` $θ -$ like' method, which consists of a backward Euler method using a $θ τ_{n}$ --time step and a forward Euler method using a $(1 - θ) τ_{n}$ --time step. The bulk of the computation is done by the backward Euler method, as the forward Euler step is equivalent to (and implemented as) a linear extrapolation. The variable $τ_{n}$ --time step integration scheme is combined with the partitioned, kinematically coupled $β -$ scheme, used to decouple the fluid and structure sub--problems. In the backward Euler step, the two sub--problems are solved in a partitioned sequential manner, and iterated until convergence. Then, the fluid and structure sub--problems are post--processed /extrapolated in the forward Euler step, and finally the $τ_{n}$ --time step is adapted. The refactorized Cauchy's one--legged ` $θ -$ like' method used in the development of the proposed method is equivalent to the midpoint rule when $θ$ = $\frac{1}{2}$ , in which case the method is non--dissipative and second--order accurate. We prove that the sub--iterative process of our algorithm is linearly convergent, and that the method is unconditionally stable when $θ \geq \frac{1}{2}$ . The numerical examples explore the properties of the method when both fixed and variable time steps are used, and in both cases shown an excellent agreement with the reference solution.

Fasma Diele, Angela Martiradonna, and C. Trenchea, "Stability and errors estimates of a second-order IMSP scheme", Discrete and Continuous Dynamical Systems Series S 15 (2022), no. 12, pp. 3645-3665.

Abstract. We analyze a second-order accurate implicit-symplectic (IMSP) scheme for reaction-diffusion systems modeling spatiotemporal dynamics of predator-prey populations. We prove stability and errors estimates of the semi- discrete-in-time approximations, under positivity assumptions. The numerical simulations confirm the theoretically derived rates of convergence and show an improved accuracy in the second-order IMSP in comparison with the first-order IMSP, at same computational cost.

John Burkardt, Wenlong Pei, and C. Trenchea, "A stress test for the midpoint time-stepping method", International Journal of Numerical Analysis and Modeling 19 (2022), no. 2-3, 299–314.

Abstract. The midpoint method can be implemented as a sequence of Backward Euler and Forward Euler solves with half time steps, allowing for improved performance of existing solvers for PDEs. We highlight the advantages of this refactorization by considering some specifics of implementation, conservation, error estimation, adaptivity, stability, and performance on several test problems.

William Layton, Wenlong Pei, and C. Trenchea, "Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna", Applied Mathematics Letters 125 (2022), 107789.

Abstract. The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional $G$ -stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method is equivalent to the backward Euler method with pre- and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.

Aziz Takhirov, C. Trenchea, and Jiajia Waters, "Second-order efficient nonlinear filter stabilization for high Reynolds number flows", Numerical Methods for Partial Differential Equations, (2021).

Abstract. In this work, we develop a second-order nonlinear filter based stabilization scheme for high Reynolds number flows. We prove the unconditional stability of the method, establish the second order consistency and discuss the dynamical tuning of the relaxation parameter. The scheme is then validated against experimental data for an isothermal turbulent flow in a Staggered Tube Bundle at Reynolds number of 18000. Numerical results are found to be in an overall good qualitative and quantitative agreement with the benchmark results.

Martina Bukač, Anyastassia Seboldt and C. Trenchea, "Refactorization of Cauchy's method: a second-order partitioned method for fluid-thick structure interaction problems", Journal of Mathematical Fluid Mechanics 23 (2021), no. 3, 64.

Abstract. This work focuses on the derivation and the analysis of a novel, strongly-coupled partitioned method for fluid-structure interaction problems. The flow is assumed to be viscous and incompressible, and the structure is modeled using linear elastodynamic equations. We assume that the structure is thick, i.e., described in the same dimension as fluid. Our newly developed numerical method is based on generalized Robin boundary conditions, as well as on the refactorization of the Cauchy's one-legged ` $θ$ -like' method, written as a sequence of Backward Euler - Forward Euler problems. This family of methods, parametrized by $θ$ , is B-stable for any $θ \in [\frac{1}{2}, 1]$ and second-order accurate for $θ = \frac{1}{2} + O (τ)$ , where $τ$ is the time step. In the proposed algorithm, the fluid and structure sub-problems, discretized using the Backward Euler scheme, are first solved iteratively until convergence. Then, the variables are linearly extrapolated, equivalent to solving Forward Euler problems. We prove that the iterative procedure is convergent, and that the pro- posed method is stable provided $θ \in [\frac{1}{2}, 1]$ . Numerical examples explore convergence rates using different values of parameters in the problem, and compare our method to other strongly-coupled partitioned schemes from the literature. We also compare our method to both a monolithic and a non-iterative partitioned solver on a benchmark problem with parameters within the physiological range of blood flow, obtaining an excellent agreement with the monolithic scheme.

John Burkardt and C. Trenchea, "Refactorization of the midpoint rule", Applied Mathematics Letters 107 (2020), 106438.

Abstract. An alternative formulation of the midpoint method is employed to analyze its advantages as an implicit second-order absolutely stable timestepping method. Legacy codes originally using the backward Euler method can be upgraded to this method by inserting a single line of new code. We show that the midpoint method, and a theta-like generalization, are B-stable. We outline three estimates of local truncation error that allow adaptive time-stepping.

William Layton, Wenlong Pei, Yi Qin, and C. Trenchea, "Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow", Numerical Methods for Partial Differential Equations, 38 (2022), no. 6, 1713–1737.

Abstract. The two-step time discretization proposed by Dahlquist, Liniger and Nevanlinna is variable step $G$ -stable. (In contrast, for increasing time steps, the BDF2 method loses A-stability and suffers non-physical energy growth in the approximate solution.) While unexplored, it is thus ideal for time accurate approximation of the Navier-Stokes equations. This report presents an analysis, for variable time-steps, of the method's stability and convergence rates when applied to the NSE. It is proven that the method is variable step, unconditionally, long time stable and second order accurate. Variable step error estimates are also proven. The results are supported by several numerical tests.

Martina Bukač and C. Trenchea, "BOundary Update via Resolvent for fluid-structure interaction", Journal of Numerical Mathematics 29 (2021), no. 1, 1--22.

Abstract. We propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous, incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler - Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second- order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm the theoretical convergence analysis results and show an excellent agreement of the proposed partitioned scheme with the monolithic scheme.

C. Trenchea, "Partitioned conservative, variable step, second-order method for magneto-hydrodynamics in Elsässer variables", ROMAI J., 15 (2019), no. 2, 117-137.

Abstract. The magnetohydrodynamics flows are governed by the Navier-Stokes equations coupled with the Maxwell equations. We propose a partitioned, variable step, second-order in time, method for the evolutionary full MHD equations, at high magnetic Reynolds number. The method is based on the refactorization of the midpoint rule. We prove the convergence of the subiterates, the energy equality at the discrete time levels, and the conservation of energy, cross-helicity and magnetic helicity.

Vince Ervin, Michaela Kubacki, William Layton, Marina Moraiti, Zhiyong Si, and C. Trenchea, "Partitioned penalty methods for the evolutionary Stokes-Darcy-transport problem", Numerical Methods for Partial Differential Equations, 35 (2019), no. 1, 349-374.

Abstract. There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algorithm development for the evolutionary transport problem has, however, focused on a quasi-static Stokes-Darcy model and a single domain (fully coupled) formulation of the transport equation. This report presents a numerical analysis of a partitioned method for contaminant transport for the fully evolutionary system. The algorithms studied are unconditionally stable with one subdomain solve per step.

Yong Li, C. Trenchea, "Existence and ergodicity for the two-dimensional stochastic Boussinesq equation", International Journal of Numerical Analysis and Modeling 15 (2018), no. 4-5, 715-728.

Abstract. The existence of solutions to the Boussinesq system driven by random exterior forcing terms both in the velocity field and the temperature is proven using a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods.

Martina Bukač, Oyekola Oyekole, and C. Trenchea, "A second-order in time approximation of fluid-structure interaction problem", SIAM J. Numer. Anal. 56 (2018), no. 1, 590-613.

Abstract. We propose and analyze a novel, second order in time, partitioned method for the interaction between an incompressible, viscous fluid and a thin, elastic structure. The proposed numerical method is based on the Crank-Nicolson discretization scheme, which is used to decouple the system into a fluid sub-problem and a structure subproblem. The scheme is loosely coupled, and therefore at every time step, each sub-problem is solved only once. Energy and error estimates for a fully discretized scheme using finite element spatial discretization are derived. We prove that the scheme is stable under a CFL condition, second-order convergent in time and optimally convergent in space. Numerical examples support the theoretically obtained results and demonstrate long time stability on a realistic example of blood flow.

Ahmet Guzel, C. Trenchea, "The Williams step increases the stability and accuracy of the hoRA time filter", Appl. Numer. Math. 131 (2018), 158-173.

Abstract. The explicit weakly-stable second-order accurate leapfrog scheme is widely used in the numerical models of weather and climate, in conjunction with the Robert-Asselin (RA) and Robert-Asselin-Williams (RAW) time filters. The RA and RAW filters successfully suppress the spurious computational mode associated with the leapfrog method, but also weakly damp the physical mode and degrade the numerical accuracy to first-order. The recent higher-order Robert-Asselin (hoRA) time filter reduces the undesired numerical damping of the RA and RAW filters and increases the accuracy to second up-to third-order. We prove that the combination of leapfrog-hoRA and Williams' step increases the stability by 25%, improves the accuracy of the amplitude of the physical mode up-to two significant digits, effectively suppresses the computational modes, and further diminishes the numerical damping of the hoRA filter.

Yong Li and C. Trenchea, "Partitioned second order method for magnetohydrodynamics in Elsässer variables", Discrete & Continuous Dynamical Systems - Ser. B 23 (2018), no. 7, 2803-2823.

Abstract. Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost. This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsässer variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Fasma Diele, Marcus R. Garvie, and C. Trenchea, "Numerical analysis of a first-order in time implicit-symplectic scheme for predator-prey systems", Computers and Mathematics with Applications 74 (2017), no. 5, 948-961.

Abstract. The numerical solution of reaction diffusion systems modelling predator-prey dynamics using implicit-symplectic (IMSP) schemes is relatively new. When applied to problems with chaotic dynamics they perform well, both in terms of computational effort and accuracy. However, until the current paper, a rigorous numerical analysis was lacking. We analyze the semi-discrete in time approximations of a first-order IMSP scheme applied to spatially extended predator-prey systems. We rigorously establish semi-discrete a priori bounds that guarantees positive and stable solutions, and prove an optimal a priori error estimate. This analysis is an improvement on previous theoretical results using standard implicit-explicit (IMEX) schemes. The theoretical results are illustrated via numerical experiments in one and two space dimensions using fully-discrete finite element approximations.

Yong Li, William Layton, C. Trenchea, "Recent developments in IMEX methods with time filters for systems of evolution equations", Journal of Computational and Applied Mathematics 299 (2016), 50-67.

Abstract. This report presents a summary of the numerical analysis of time filters used to control the unstable mode in the Crank-Nicolson-Leapfrog discretization of evolution equations.

Costica Morosanu, Silviu Paval, and C.Trenchea, " Analysis of stability and error estimates for three methods approximating a nonlinear reaction-diffusion equation", J. Appl. Anal. Comput. 7 (2017), no. 1, 1-19.

Abstract. We present the error analysis of three time-stepping schemes used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition. We prove $L^{1} (L^{1})$ stability by maximum principle arguments, and derive error estimates using energy methods for the implicit Euler, and two implicit-explicit approaches, a linearized scheme and a fractional step method. A numerical experiment validates the theoretical results, comparing the accuracy of the methods.

Nan Jiang, Muhammad Mohebujjaman, Leo G. Rebholz, and C.Trenchea, " An optimally accurate discrete regularization for second order timestepping methods for Navier-Stokes equations". Comput. Methods Appl. Mech. Engrg. 310 (2016), 388-405.

Abstract. We propose a new numerical regularization for finite element spatial discretization of the Navier-Stokes equations (NSE), a family of implicit-explicit (IMEX) second order timestepping schemes. The method combines a linear treatment of the advection term and stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. Only a linear Oseen problem needs to be solved at each timestep. We prove that the methods are unconditionally stable and second order convergent. Numerical examples verify the convergence rate and show the stabilization term clearly improves the stability of the tested flow.

Vincent J. Ervin, Michaela Kubacki, William Layton, Marina Moraiti, Zhiyong Si, C. Trenchea, "On Limiting Behavior of Contaminant Transport Models in Coupled Surface and Groundwater Flows", Axioms 4 (2015), 518-529.

Abstract. There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The analysis of models including the transport of contaminants has, however, focused on a quasi-static Stokes-Darcy model. Herein we consider the fully evolutionary system including contaminant transport and analyze its quasi-static limits.

Yong Li, C. Trenchea, "Analysis of time filters used with the leapfrog scheme", Proceedings of the VI Conference on Computational Methods for Coupled Problems in Science and Engineering, Venice, Italy, May 18-20, 2015.

Abstract. We present the linear analysis of recent time filters used in numerical weather prediction. We focus on the accuracy and the stability of the leapfrog scheme combined with the Robert-Asselin-Williams filter, the higher-order Robert-Asselin type time filter, the composite-tendency Robert-Asselin-Williams filter and a more discriminating filter.

Martina Bukač, William Layton, C.Trenchea, Marina Moraiti, and Hoang Tran, "Analysis of partitioned methods for Biot system", Numerical Methods for Partial Differential Equations, 31 (2015), no. 6, 1769-1813.

Abstract. In this work, we present a comprehensive study of several partitioned methods for the coupling of flow and mechanics. We derive energy estimates for each method for the fully discrete problem. We write the obtained stability conditions in terms of a key control parameter defined as a ratio of the coupling strength and the speed of propagation. Depending on the parameters in the problem, give the choice of the partitioned method which allows the largest time step.

Nicholas Wilson, Alexander Labovsky, C. Trenchea, "High accuracy method for magnetohydrodynamics system in Elsässer variables", Comput. Methods Appl. Math. 15 (2015), no. 1, 97-110.

Abstract. A method has been developed recently by the third author, that allows for decoupling of the evolutionary full MagnetoHydroDynamics (MHD) system in the Elsässer variables. The method entails the implicit discretization of the subproblem terms and the explicit discretization of coupling terms, and was proven to be unconditionally stable. In this paper we build on that result by introducing a high-order accurate deferred correction method, which also decouples the MHD system. We perform the full numerical analysis of the method, proving the unconditional stability and second order accuracy of the two-step method. We also use a test problem to verify numerically the claimed convergence rate.

Yong Li, C. Trenchea, "A higher-order Robert-Asselin type time filter", J. Comp. Phys. 259C (2014), 23--32.

Abstract. The Robert-Asselin (RA) time filter combined with leapfrog scheme is widely used in numerical models of weather and climate. It successfully suppresses the spurious computational mode associated with the leapfrog method, but it also weakly dampens the physical mode and degrades the numerical accuracy. The Robert-Asselin-Williams (RAW) time filter is a modification of the RA filter that reduces the undesired numerical damping of RA filter and increases the accuracy. We propose a higher-order RA (hoRA) type time filter which effectively suppresses the computational modes and achieves third-order accuracy with the same storage requirement as RAW filter. Like RA and RAW filters, the hoRA filter is non-intrusive, and so it would be easily implementable. The leapfrog scheme with hoRA filter is almost as accurate, stable and efficient as the intrusive third-order Adams-Bashforth (AB3) method.

Keith Galvin, Leo G. Rebholz, and C.Trenchea, "Efficient, unconditionally stable, and optimally accurate FE algorithms for approximate deconvolution models", SIAM J. Numer. Anal. 52 (2014), no. 2, 678-707.

Abstract. This paper addresses an open question of how to devise numerical schemes for approximate deconvolution fluid flow models that are efficient, unconditionally stable, and optimally accurate. We propose, analyze and test a scheme for these models that has each of these properties for the case of homogeneous Dirichlet velocity boundary conditions. There are several important components to the derivation, both at the continuous and discrete levels, which allow for these properties to hold. The proofs of unconditional stability and optimal convergence are carried out through the use of a special choice of test function and some technical estimates. Numerical tests are provided that confirm the effectiveness of the scheme.

Hoang A. Tran, C.Trenchea, and Clayton G. Webster, " A convergence analysis of stochastic collocation method for Navier-Stokes equations with random input data", ORNL Technical Report, Oak Ridge National Laboratory, 2014.

Abstract. Stochastic collocation method has proved to be an efficient method and been widely applied to solve various partial differential equations with random input data, including Navier-Stokes equations. However, up to now, rigorous convergence analyses are limited to linear elliptic and parabolic equations; its performance for Navier-Stokes equations was demonstrated mostly by numerical experiments. In this paper, we present an error analysis of the stochastic collocation method for a semi-implicit Backward Euler discretization for NSE and prove the exponential decay of the interpolation error in the probability space. Our analysis indicates that due to the nonlinearity, as final time $T$ increases and NSE solvers pile up, the accuracy may be reduced significantly. Subsequently, an illustrative computational test of time dependent fluid flow around a bluff body is provided.

William Layton, Hoang Tran, and C.Trenchea, " Numerical analysis of two partitioned time stepping methods for MHD flows at low magnetic Reynolds numbers", Numerical Methods for Partial Differential Equations 30 (2014), no. 4, 1083-1102.

Abstract. Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes (NSE) equations in fluid dynamics and Maxwell equations in eletromagnetism. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of non-model problems can require different meshes, time steps and methods. In most terrestrial applications, MHD flows occur at low magnetic Reynold numbers. We introduce two partitioned methods to solve evolutionary MHD equations in such cases. The methods we study allow us at each time step to call NSE and Maxwell codes separately, each possibly optimized for the subproblem's respective. Error analysis and experiments supporting the theory are given.

Nicholas Hurl, William Layton, Yong Li, C. Trenchea, "Stability Analysis of the Crank-Nicolson-Leap-Frog method with the Robert-Asselin-Williams time filter", BIT. Numerical Mathematics 54 (2014), no. 4, 1009-1021.

Abstract. Geophysical flow simulations have evolved sophisticated implicit-explicit time stepping methods (based on fast-slow wave splittings) followed by time filters to control any unstable models that result. Time filters are modular and parallel. Their effect on stability of the overall process has been tested in numerous simulations. In this paper, we study the stability of the Crank-Nicolson-Leapfrog scheme with the Robert-Asselin-Williams time filter.

C.Trenchea, "Second order implicit for local effects and explicit for nonlocal effects is unconditionally stable", ROMAI J. 12 (2016), no. 1, 163-178.

Abstract. A family of implicit-explicit second order time-stepping methods is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. The methods we consider are implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proved by the energy method and by algebraic techniques. This is the first solution to the problem of finding a scheme for (1.1) that is (provably) unconditionally stable and treats the Cu term explicitly. First order schemes were known in [2,10] and [10] gives a second order scheme stable provided all operators commute.

C.Trenchea, " Stability of partitioned IMEX methods for systems of evolution equations with skew-symmetric coupling", ROMAI J. 10 (2014), no. 2, 175-189.

Abstract. Stability is proven for an implicit-explicit, second order, two step method for uncoupling a system of two evolution equations with exactly skew symmetric coupling. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. The method proposed is an interpolation of the Crank-Nicolson Leap Frog (CNLF) combination with the BDF2-AB2 combination, being stable under the time step condition suggested by linear stability theory for the Leap-Frog scheme and BDF2-AB2.

C.Trenchea, " Unconditional Stability of a Partitioned IMEX Method for Magnetohydrodynamic Flows", Applied Mathematical Letters 27 (2014), 97-100.

Abstract. The MHD flows are governed by the Navier-Stokes equations coupled with the Maxwell equations through coupling terms. We prove the unconditional stability of a partitioned method for the evolutionary full MHD equations, at high magnetic Reynolds number, in the Elsässer variables. The method we analyze is a first order, one step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of coupling terms.

William Layton, Nathaniel Mays, Monika Neda, and C.Trenchea, " Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations", ESAIM Mathematical Modelling and Numerical Analysis M2AN 48 (2014), no. 3, 765-793.

Abstract. We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations of the Navier-Stokes equations. The method is: Step 1: Advance the NSE one time step, Step 2: Regularize to obtain the approximation at the new time level. The algorithmic key is that Step 2 is a modular regularization uncoupled from Step 1. Previous analysis of this approach has been for simple time stepping methods in Step 1 and specific regularization operators in Step 2 such as filter based stabilization. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing BDF2 time discretization in Step 1, and (ii) general (linear or nonlinear) regularization operators in Step 2. We give a complete stability analysis, derive conditions on the Step 2 regularization operator for which the combination has good stabilization effects, characterize the numerical dissipation induced by Step 2, prove an asymptotic error estimate incorporating the numerical error of the method used in Step 1 and the regularization's consistency error in Step 2 and provide numerical tests. Some tests verify the presented convergence theory and some tests are beyond the theory developed. The latter suggest several directions for further development of modular stabilization methods.

Marcus R. Garvie, and C. Trenchea, "Identification of space-time distributed parameters in the Gierer-Meinhardt reaction-diffusion systems", SIAM J. Appl. Math. 74 (2014), no. 1, 147-166.

Abstract. We consider parameter identification for the classic Gierer-Meinhardt reaction-diffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39] was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology. In our paper the parameters are extended in time and space and used as distributed control variables. The methodology employs PDE-constrained optimization in the context of image-driven spatiotemporal pattern formation. We prove the existence of optimal solutions, derive an optimality system, and determine optimal solutions. The results of numerical experiments in 2D are presented using the finite element method, which illustrates the convergence of a variable-step gradient algorithm for finding the optimal parameters of the system. A practical target function was constructed for the optimal control algorithm corresponding to the actual image of a marine angelfish.

Marcus R. Garvie, and C. Trenchea, "A three level finite element approximation of a pattern formation model in developmental biology", Numer. Math. 127 (2014), no. 3, 397-422.

Abstract. This paper concerns a second-order, three level piecewise linear finite element scheme 2-SBDF [J. RUUTH, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), pp. 148-176] for approximating the stationary (Turing) patterns of a well-known experimental substrate-inhibition reaction-diffusion (`Thomas') system [D. THOMAS, Artificial enzyme membranes, transport, memory and oscillatory phenomena, in Analysis and control of immobilized enzyme systems, D. Thomas and J.P. Kernevez, eds., Springer, 1975, pp. 115-150]. A numerical analysis of the semi-discrete in time approximations leads to semi-discrete a priori bounds and an optimal error estimate. The analysis highlights the technical challenges in undertaking the numerical analysis of multi-level ( $\geq 3$ ) schemes. We illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology, namely, to approximate the Turing patterns of the Thomas system over a schematic mammal skin domain with fixed geometry at various scales. We also make some comments on the correct procedure for simulating Turing patterns in general reaction-diffusion systems.

William Layton, Hoang Tran and C.Trenchea, "Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater - surface water flows", SIAM J. Numer. Anal., 51 (2013), no. 1, 248-272.

Abstract. The most effective simulations of the multi-physics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the sub-physics problems, have recently been studied for the Stokes-Darcy coupling with convergence established over bounded time intervals (with constants growing exponentially in t). This report analyzes and tests two such partitioned (non-iterative, domain decomposition) methods for the fully evolutionary Stokes-Darcy problem. Under a modest time step restriction of the form $C = C$ (physical parameters) we prove unconditional asymptotic (over $0 \leq t < \infty$ stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over $0 \leq t < \infty$ .

William Layton, Hoang Tran, and C.Trenchea, "Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds number", in Recent Advances in Scientific Computing and Applications: Proceedings of the 8th International Conference on Scientific Computing and Applications, Eds. Jichun Li, Eric Macharro, and Hongtao Yang, AMS Contemporary Mathematics, vol. 586, 2013.

Abstract. MHD flows are governed by the Navier-Stokes equations coupled with the Maxwell equations. Broadly, MHD flows in astrophysics occur at large magnetic Reynolds numbers while those in terrestrial applications, such as liquid metals, occur at small magnetic Reynolds numbers, the case considered herein. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of non-model problems can require different meshes, time steps and methods. We introduce implicit-explicit (IMEX) methods where the MHD equations can be evolved in time by calls to the NSE and Maxwell codes, each possibly optimized for the subproblem's respective physics.

William Layton and C.Trenchea, "Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations", Applied Numerical Mathematics, 62:2 (2012) 112-120.

Abstract. Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap Frog (CNLF) combination and the BDF2-AB2 combination. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. For CNLF we prove stability for the coupled system under the time step condition suggested by linear stability theory for the Leap-Frog scheme. This seems to be a first proof of a widely believed result. For BDF2-AB2 we prove stability under a condition that is better than the one suggested by linear stability theory for the individual methods.

Abigail Bowers, Leo G. Rebholz, Aziz Takhirov and C.Trenchea, "Improved accuracy in regularization models of incompressible flow via adaptive nonlinear filtering", Internat. J. Numer. Methods Fluids, 70 (2012), no. 7, 805-828.

Abstract. We study adaptive nonlinear filtering in the Leray regularization model for incompressible, viscous Newtonian flow. The filtering radius is locally adjusted so that resolved flow regions and coherent flow structures are not `filtered-out', which is a common problem with these types of models. A numerical method is proposed that is unconditionally stable with respect to timestep, and decouples the problem so that the filtering becomes linear at each timestep and is decoupled from the system. Several numerical examples are given that demonstrate the effectiveness of the method.

William Layton, Iuliana Stanculescu and C.Trenchea, "Theory of NS-

\overset{―}{ω}

model: a compliment to the NS-

α

model", Commun. Pure Appl. Anal., 10 (2011), 1763-1777.

Abstract. We study a new regularization of the Navier-Stokes equations, the NS- $\overset{―}{ω}$ model. This model has similarities to the NS- $α$ model, but its structure is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo a subsequence) to a weak solution of the Navier-Stokes equations as the averaging radius decreases to zero. We then apply turbulence phenomenology to the model to obtain insight into its predictions.

William Layton, Leo G. Rebholz and C.Trenchea, "Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow", J. Math. Fluid Mech., (2011), 1-30.

Abstract. Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear and yet current models for it use linear filters to select the eddies that will be damped. In this report we consider for the first time nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow problems. The particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to calculating a linear filter of similar form. We then analyze nonlinear filter based stabilization for the Navier-Stokes equations. We give a precise analysis of the numerical diffusion and error in this process.

William Layton, and C.Trenchea, "The Das-Moser commutator closure for filtering through a boundary is well posed", Mathematical and Computer Modelling, 53 (2011), 566-573.

Abstract. When filtering through a wall with constant averaging radius, in addition to the subfilter scale stresses, a non-closed commutator term arises. We consider a proposal of Das and Moser to close the commutator error term by embedding it in an optimization probem. This report shows that this optimization based closure, with a small modification, leads to a well posed problem showing existence of a minimizer. We also derive the associated first order optimality conditions.

Alexander Labovsky and C.Trenchea, "Large eddy simulation for turbulent MHD flows", J. Math. Anal. Appl. vol. 377, no. 2 (2011) 516-533.

Abstract. We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvèn waves, with the velocity converging to that of the MHD, as $δ_{1}$ , $δ_{2}$ tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD.

Marcus R. Garvie, Philip K. Maini, and C.Trenchea, "An efficient and robust numerical algorithm for estimating parameters in Turing systems", J. Comp. Phys. 229 (2010) 7058-7071.

Abstract. We present a new algorithm for estimating parameters in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.

William Layton, Mike Sussman, and C.Trenchea, "Bounds on energy, magnetic helicity and cross helicity dissipation rates of approximate deconvolution models of turbulence for MHD flows", Numer. Funct. Anal. Optim., 31:5, 577-595 (2010).

Abstract. We consider a family of high accuracy, approximate deconvolution models of turbulent magnetohydrodynamic flows. For body force driven turbulence, we prove directly from the model's equations of motion the following bounds on the model's time averaged energy dissipation rate, time averaged cross helicity dissipation rate and magnetic helicity dissipation rate, where U, B, L are the global velocity scale, global magnetic field scale and length scale, R is a dimensionless constant related to fluid and magnetic Reynolds numbers, S is the reciprocal of the product of fluid density times free-space permeability and $δ$ is the LES filter radius.

Alexander Labovsky and C.Trenchea, "Approximate Deconvolution Models for MagnetoHydroDynamics", Numer. Funct. Anal. Optim., 31:12, 1362-1385 (2010).

Abstract. We consider the family of approximate deconvolution models (ADM) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove the existence and uniqueness of solutions to the ADM-MHD equations, their weak converge to the solution of the MHD equations as the averaging radii tend to zero, and derive a bound on the modeling error. We demonstrate that the energy and helicity of the models are conserved, and the models preserve the Alfvèn waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models.

Marcus R. Garvie and C.Trenchea, "Spatiotemporal dynamics of two generic predator-prey models", J. Biol. Dyn.vol. 4, No. 6 (2010) 559-570.

Abstract. We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an $L^{\infty}$ -stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori $L^{\infty}$ -uniform bounds, given any nonnegative $L^{\infty}$ -estimate to general reaction-diffusion systems is discussed, how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be `trapped' in an invariant region of phase space.

Max D. Gunzburger, Eunjung Lee, Yuki Saka, C. Trenchea and Xiaoming Wang, "Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence", J. Sci. Comput., 45:1-3 (2010) 294-332.

Abstract. Fluid turbulence is usually characterized by the Navier-Stokes equations with a large Reynolds number. The simulation of turbulence model is known to be very difficult. In this paper, we use an artificial spectral viscosity to make the simulation of turbulence tractable. The model introduce various parameters and we pose a question whether an effective choice of a parameter can be made using the mathematical analysis. We show that the resulting partial differential equation is well-posed and its consistency. Then, we consider a semi-implicit discretization of the equation and investigate the stability.

A. Labovschii, W. Layton, C. Manica , M. Neda, Leo G. Rebholz, I. Stanculescu, and C. Trenchea, "Mathematical Architecture of Approximate Deconvolution Models of Turbulence", Quality and Reliability of Large-Eddy Simulations, Editors: Johan Meyers, Bernard Geurts and Pierre Sagaut, ERCOFTAC Series, Volume 12, p. 3-20, Springer, Berlin, 2008.

Abstract. This report presents the mathematical foundation of approximate deconvolution LES models together with the model phenomenology downstream of the theory.

Max D. Gunzburger, Janet Peterson, and C. Trenchea, "The velocity and magnetic field tracking problem for MHD flows with distributed controls", Int. J. Pure Appl. Math. 42 (2007), no. 2, 289-296.

Abstract. We consider the mathematical formulation and the analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluid in a bounded two-dimensional domain through the adjustment of distributed controls. Existence of optimal solutions is proved and first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown.

Marcus R. Garvie and C. Trenchea, "Finite element approximations of spatially extended predator-prey interactions with the Holling type II functional response", Numer. Math. 107 (2007), no. 4, 641-667.

Abstract. We present the numerical analysis of two well-known reaction-diffusion systems modeling predator-prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. Results are presented for two fully-practical piecewise linear finite element methods. We establish a priori estimates and error bounds for the semi-discrete and fully-discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena (e.g., spiral waves and chaos) are presented in 1-D and 2-D.

Max D. Gunzburger and C. Trenchea, "Analysis of optimal control problem for three-dimensional coupled modified Navier-Stokes and Maxwell equations", J. Math. Anal. Appl., no. 333 (2007), 295-310.

Abstract. We consider the mathematical formulation and the analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain through the adjustment of distributed controls. The existence of optimal solutions is shown, the Gateux differentiability for the MHD system with respect to controls is proved, and the optimality system is obtained.

Marcus R. Garvie and C. Trenchea, " Optimal control of a nonlinear `fish-phytoplankton-zooplankton' system", SIAM J. Control Optim., 46 (2007), 775-791.

Abstract. We consider the mathematical formulation, analysis, and numerical solution of an optimal control problem for a nonlinear `nutrient-phytoplankton-zooplankton-fish' reaction-diffusion system. We study the existence of optimal solutions, derive an optimality system, and determine optimal solutions. In the original spatially homogeneous formulation the dynamics of plankton were investigated as a function of parameters for nutrient levels and fish predation rate on zooplankton. In our paper the model is spatially extended and the parameter for fish predation treated as a multiplicative control variable. The model has implications for the biomanipulation of food-webs in eutrophic lakes to help improve water quality. In order to illustrate the control of irregular spatiotemporal dynamics of plankton in the model we implement a semi-implicit (in time) finite element method with `mass lumping', and present the results of numerical experiments in two space dimensions.

Marcus R. Garvie and C. Trenchea, " Biomanipulation of Food-Webs in Eutrophic Lakes", Proceedings of the 46th IEEE Conference on Decision and Control, p. 3460-3465, New Orleans, LA, USA, Dec. 12-14, 2007.

Max D. Gunzburger and C. Trenchea, "Optimal control of time-periodic MHD equations ", Nonlinear Anal., Theory Methods Appl., 63 (2005), no. 5-7, e1687-e1699, Invited Talks from the Fourth World Congress of Nonlinear Analysts (WCNA 2004).

Max D. Gunzburger and C. Trenchea, "Analysis and discretization of an optimal control problem for the time-periodic MHD equations ", J. Math Anal. Appl. 308(2005), no. 2, 440-466.

Abstract. We consider the mathematical formulation and analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluid in a bounded two-dimensional domain through the adjustment of distributed controls. Existence of optimal solutions is proved and first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time approximations are defined and their convergence to the exact optimal solutions is shown.

C. Trenchea, "Periodic optimal control of the Boussinesq equation ", Nonlinear Anal., Theory Methods Appl., 53A, No.1, 81-96 (2003).

Abstract. This paper is concerned with the existence and the maximum principle for the optimal control problem governed by the Boussinesq equation. The case of internal controllers is examined.

C. Trenchea, "Periodic optimal control of the Euler-Bernoulli equation ", Commun. Appl. Anal 7 (2003), no. 1, 115--125.

Abstract. This paper is concerned with the existence and the maximum principle for the optimal control problems governed by the periodic www-Bernoulli equation in one dimension with internal controllers.

Marc Quincampoix and C. Trenchea, "Hamilton-Jacobi equation and optimality conditions for control systems governed by semilinear parabolic equations with boundary control", tech. rep., UniversitÈ de Bretagne Occidentale, 2003.

Abstract. We characterize the value function by an appropriate Hamilton Jacobi Bellman equation (in the viscosity sense) and derive optimality conditions from the knowledge of the value function.

C. Trenchea, "Optimal control of an elliptic equation under periodic conditions", Mem. Sect. Stiint. Acad. Romana Ser. IV 25 (2002), 23--35 (2005).

Abstract. We find explicitly the optimal control for an elliptic equation with respect to two different cost functionals.

Thomas R. Bewley and C. Trenchea , "Noncooperative optimization of controls for time periodic Navier-Stokes systems with multiple solutions", AIAA 2002-2754 . (2002)

Abstract. Time-periodic systems governed by differential equations are somewhat difficult to consider in the numerical setting because they may possess many solutions. The number of solutions of such systems may be finite or infinite. Further, some trajectories which are exactly time-periodic over a given period might only approximately solve the governing equation, whereas nearby trajectories which exactly solve the governing equation might only be approximately time-periodic over the given period. The difficulty of the time-periodic setting is compounded in the case of systems governed by the Navier-Stokes equation, as the solutions of such systems in the time-evolving setting may be chaotic and multiscale. When considering the optimization of controls for such systems in the time-periodic setting, the situation is thus particularly delicate, as one doesn't know a priori which time-periodic solution (or approximate solution) one should design the controls for. The present brief note motivates this work, presents the structure of our analysis, and outlines the resulting numerical algorithm.

Costica Morosanu and C. Trenchea, "Identification for nonlinear periodic wave equation", Appl. Math. Optimization, 44 (2001), 87--104.

Abstract. This work is concerned with an approximation process for the identification of nonlinearities in the nonlinear periodic wave equation. It is based on the least-squares approach and on a splitting method. A numerical algorithm of gradient type and the numerical implementation are given.

C. Trenchea, "Optimal control of the periodic string equation with internal control ", J. Optimization Theory Appl. 101(1999), 429--447.

Abstract. This paper is concerned with the existence and the maximum principle for the optimal control problem governed by the periodic vibrating string equation with Dirichlet boundary conditions. The case of internal controllers is examined.


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