A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

Amiya K Pani,A S Vasudeva Murthy,Vidar Thomée

doi:10.1515/cmam-2020-0009

Abstract

Abstract We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m ≥ 1 {m\geq 1} intervals of length k m {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form C 0 ⁢ h 2 + C m ⁢ k {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where C m ≤ C ′ + C ′′ m {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}} . This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

Highlights

We shall consider the numerical solution by finite differences and splitting of the convectiondiffusion equation
In the method that we study in this paper, we begin by discretizing (1.1) in the spatial variables
The basis of the method we study here, will be the splitting method analogous to (1.4), (1.5), defined by un = Ek un−1 + kf n for n ≥ 1, with u0 = v, where Ek = ekB e−kA ≈ E(k) = e−k(A−B)

Summary

Introduction

We shall consider the numerical solution by finite differences and splitting of the convectiondiffusion equation. Note that we use capital letters for functions in Ω and lowercase letters for vectors in Ph. We consider the second-order spatially discrete, continuous in time, finite difference ODE system for u(t) ∈ Ph, du dt. This method, which we will refer to as our fully discrete method, replaces at each time step the backward Euler solution of our nonsymmetric problem (1.7) by a backward Euler approximation of a symmetric parabolic problem, followed by an explicit finite difference solution of a hyperbolic problem. After the introduction of notation and some preliminary observations, the spatially semidiscrete problem (1.7), (1.8), the backward Euler method (1.9) and the corresponding basic time splitting method (1.10) will be analyzed, where we will show O(h2) and O(h2 + k) error bounds for these methods, respectively. 5, we illustrate our theoretical results with computations for examples with one and two spatial dimensions

Notation and Preliminaries

The Fully Discrete Splitting Method

Numerical Illustrations