Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which also contains numerical experiments. Version 3 and 4: Only comments added. Version 2, page 2: Clarified statement on stability issues for ADI schemes with more than two operator

An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians

We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac and the Hopf-Lax formulae. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation.Comment: 24 page

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

Difference schemes of arbitrary order of accuracy for semilinear parabolic equations

The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T1(t) + T2(t), ∂u/∂x ≠ = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)Comment: This paper has been accepted for publication by IEEE Trans. on Automatic Control, and is available at http://dx.doi.org/10.1109/TAC.2018.2880160. arXiv admin note: substantial text overlap with arXiv:1710.0991

Spatial and Physical Splittings of Semilinear Parabolic Problems

Splitting methods are widely used temporal approximation schemes for parabolic partial differential equations (PDEs). These schemes may be very efficient when a problem can be naturally decomposed into multiple parts. In this thesis, splitting methods are analysed when applied to spatial splittings (partitions of the computational domain) and physical splittings (separations of physical processes) of semilinear parabolic problems. The thesis is organized into three major themes: optimal convergence order analysis, spatial splittings and a physical splitting application.In view of the first theme, temporal semi-discretizations based on splitting methods are considered. An analysis is performed which yields convergence without order under weak regularity assumptions on the solution, and convergence orders ranging up to classical for progressively more regular solutions. The analysis is performed in the framework of maximal dissipative operators, which includes a large number of parabolic problems. The temporal results are also combined with convergence studies of spatial discretizations to prove simultaneous space–time convergence orders for full discretizations.For the second theme, two spatial splitting formulations are considered. For dimension splittings each part of the formulation represents the evolution in one spatial dimension only. Thereby, multidimensional problems can be reduced to families of one-dimensional problems. For domain decomposition splittings each part represents a problem on only a smaller subdomain of the full domain of the PDE. The results of the first theme are applied to prove optimal convergence orders for splitting schemes used in conjunction with these two splitting formulations. The last theme concerns the evaluation of a physical splitting procedure in an interdisciplinary application. A model for axonal growth out of nerve cells is considered. This model features several challenges to be addressed by a successful numerical method. It consists of a linear PDE coupled to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. The biological model parameters imply a wide range of scales, both in time and space. Based on a physical splitting, a tailored scheme for this model is constructed. Its robustness and efficiency are then verified by numerical experiments

Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES

© 2018 Society for Industrial and Applied Mathematics. A procedure for the numerical approximation of high-dimensional Hamilton–Jacobi–Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen

Semi-Lagrangian methods for parabolic problems in divergence form

Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches