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A simple predictor-corrector scheme without order reduction for advection-diffusion-reaction problems

Thi Tam Dang, Lukas Einkemmer, Alexander Ostermann

TL;DR

The paper tackles order reduction in splitting methods for semilinear advection–diffusion–reaction PDEs with nonlinear dependence on $u$ and $\nabla u$ under inhomogeneous Dirichlet data. It introduces a boundary-corrected predictor–corrector scheme: a diffusion predictor using $f(u_n,\nabla u_n)$ and a simple explicit Euler corrector for the advection–reaction part, achieving second-order accuracy under mild regularity. The authors prove first-order convergence for the base scheme and second-order convergence for the predictor–corrector variant (up to a logarithmic factor) using analytic semigroup theory and careful local-error analysis, complemented by numerical experiments in 1D that confirm the predicted rates and demonstrate substantial improvements over classical Lie/Strang splitting in the presence of nontrivial boundary conditions. The work provides a practical, efficient approach to solving ADR problems without order reduction, suitable for nonlinear gradient-dependent fluxes and time-dependent boundaries.

Abstract

Treating diffusion and advection/reaction separately is an effective strategy for solving semilinear advection-diffusion-reaction equations. However, such an approach is prone to suffer from order reduction, especially in the presence of inhomogeneous Dirichlet boundary conditions. In this paper, we extend an approach of Einkemmer and Ostermann [SIAM J. Sci. Comput. 37, A1577-A1592, 2015] to advection-diffusion-reaction problems, where the advection and reaction terms depend nonlinearly on both the solution and its gradient. Starting from a modified splitting method, we construct a predictor-corrector scheme that avoids order reduction and significantly improves accuracy. The predictor only requires the solution of a linear diffusion equation, while the corrector is simply an explicit Euler step of an advection-reaction equation. Under appropriate regularity assumptions on the exact solution, we rigorously establish second-order convergence for this scheme. Numerical experiments are presented to confirm the theoretical results.

A simple predictor-corrector scheme without order reduction for advection-diffusion-reaction problems

TL;DR

The paper tackles order reduction in splitting methods for semilinear advection–diffusion–reaction PDEs with nonlinear dependence on and under inhomogeneous Dirichlet data. It introduces a boundary-corrected predictor–corrector scheme: a diffusion predictor using and a simple explicit Euler corrector for the advection–reaction part, achieving second-order accuracy under mild regularity. The authors prove first-order convergence for the base scheme and second-order convergence for the predictor–corrector variant (up to a logarithmic factor) using analytic semigroup theory and careful local-error analysis, complemented by numerical experiments in 1D that confirm the predicted rates and demonstrate substantial improvements over classical Lie/Strang splitting in the presence of nontrivial boundary conditions. The work provides a practical, efficient approach to solving ADR problems without order reduction, suitable for nonlinear gradient-dependent fluxes and time-dependent boundaries.

Abstract

Treating diffusion and advection/reaction separately is an effective strategy for solving semilinear advection-diffusion-reaction equations. However, such an approach is prone to suffer from order reduction, especially in the presence of inhomogeneous Dirichlet boundary conditions. In this paper, we extend an approach of Einkemmer and Ostermann [SIAM J. Sci. Comput. 37, A1577-A1592, 2015] to advection-diffusion-reaction problems, where the advection and reaction terms depend nonlinearly on both the solution and its gradient. Starting from a modified splitting method, we construct a predictor-corrector scheme that avoids order reduction and significantly improves accuracy. The predictor only requires the solution of a linear diffusion equation, while the corrector is simply an explicit Euler step of an advection-reaction equation. Under appropriate regularity assumptions on the exact solution, we rigorously establish second-order convergence for this scheme. Numerical experiments are presented to confirm the theoretical results.

Paper Structure

This paper contains 10 sections, 2 theorems, 43 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model problem and proposed predictor-corrector scheme
  3. Convergence analysis for the first-order scheme
  4. Analytical framework
  5. Local error
  6. Global error
  7. Convergence analysis for the second-order predictor-corrector scheme
  8. Local error
  9. Global error
  10. Numerical examples

Key Result

Theorem 2.1

Let Assumption as1 be satisfied. Then, there exists a constant $\tau_0>0$ such that, for all step sizes satisfying $0<\tau \le \tau_0$, the scheme in Algorithm alg 1 satisfies the global error estimate where the constant $C$ depends on $T$ but is independent of $\tau$ and $n$.

Figures (1)

  • Figure 1: We solve a one-dimensional diffusion-reaction equation with different nonlinearities and various boundary conditions. The absolute error at $t = 0.5$ is computed by comparing the numerical solution with a reference solution. This error is measured in a discrete $H^{2}$ norm.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Example 5.4