Papers:
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.
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 Elsasser variables. The method we analyze is a rst order, one step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of coupling terms.
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.
Abstract. We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations of the following type. Given un, Step 1: Advance one time step to get wn+1, Step 2: Regularize and relax wn+1 to obtain un+1 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.
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 &Delta t &le C where C = C(physical parameters) we prove unconditional asymptotic (over 0 &le t < &infin) stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over 0 &le t < &infin.
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.
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.
Abstract. We study a new regularization of the Navier-Stokes equations, the NS-&omega model. This model has similarities to the NS-&alpha 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.
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.
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.
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 Alfvén waves, with the velocity converging to that of the MHD, as &delta1, &delta2 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.
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.
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 eld 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 &delta is the LES filter radius.
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 Alfvén waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models.
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&infin-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori Lp-estimate, uniform in time, for all p&ge 1, implies L&infin- uniform bounds, given any nonnegative L&infin-initial data. The applicability of the L&infin-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.
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.
Abstract. This report presents the mathematical foundation of approximate deconvolution LES models together with the model phenomenology downstream of the theory.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Abstract.
In the present work, the idea of noncooperative optimization is
applied in an attempt to develop a tractable framework to solve the
problem of optimization of controls for time-periodic Navier-Stokes
systems. The noncooperative aspect of the optimization, however, is
somewhat nonstandard: the best controls are found for the worst (of
the many) time-periodic solutions of the governing equation. As the
number of solutions may be finite, we have employed a technique
of first looking at a suitable approximation
of the time-periodic system of interest with an infinite number of
solutions, finding the solution to this approximate system with a
gradient-based algorithm leveraging an adjoint analysis, then refining
the level of approximation until we have solved (with a sufficient
level of accuracy) the optimization problem we are actually interested
in.
Abstract. We find explicitly the optimal control for an elliptic equation with respect to two different cost functionals.
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.
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.
Abstract. We consider a general methodology for parameter identi cation in systems of reaction-diffusion equations. To demonstrate the method we focus on 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 space and time 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 the optimal solutions. The results of some numerical experiments in 2-D are presented using the fi nite element method, which illustrates the convergence of a variable-step gradient algorithm for fi nding the optimal parameters of the system.
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 (&ge 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.
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Last updated: June 8, 2011 by Catalin Trenchea |
