[Paper Review] Avoiding order reduction when integrating diffusion-reaction boundary value problems with exponential splitting methods

I. Alonso-Mallo, B. Cano+1 more|arXiv (Cornell University)|May 4, 2017

Numerical methods for differential equations9 references3 citations

TL;DR

This paper proposes a novel technique to prevent order reduction in exponential splitting methods when solving diffusion-reaction boundary value problems with non-homogeneous Dirichlet, Neumann, and Robin boundary conditions. By modifying the time integration process using exact boundary corrections via exponential integrators and Krylov-based matrix functions, the method restores full convergence order—proven to be first-order for Lie-Trotter and second-order for Strang splitting—under general spatial discretizations and smoothness assumptions.

ABSTRACT

In this paper, we suggest a technique to avoid order reduction in time when integrating reaction-diffusion boundary value problems under non-homogeneous boundary conditions with exponential splitting methods. More precisely, we consider Lie-Trotter and Strang splitting methods and Dirichlet, Neumann and Robin boundary conditions. Beginning from an abstract framework in Banach spaces, a thorough error analysis after full discretization is performed and some numerical results are shown which corroborate the theoretical results.

Motivation & Objective

• Address the persistent issue of order reduction in exponential splitting methods when solving nonlinear diffusion-reaction problems with non-homogeneous boundary conditions.
• Extend previous results on linear problems with non-homogeneous BCs to nonlinear reaction-diffusion systems.
• Provide a rigorous error analysis under general spatial discretizations and in the maximum norm to ensure broad applicability.
• Ensure practical implementability by deriving exact formulas for Lie-Trotter and Strang splitting methods with boundary corrections.
• Demonstrate that full convergence order is preserved in both local and global errors after full space-time discretization.

Proposed method

• Formulate the problem in an abstract Banach space framework with operators A (differential) and ∂ (boundary), allowing for general PDEs.
• Introduce a correction mechanism using the resolvent operator K(z) to handle non-homogeneous boundary data in exponential integrators.
• Apply Lie-Trotter and Strang splitting to the abstract problem u' = Au + f(t,u), with boundary condition ∂u = g(t), and derive modified time-stepping formulas.
• Use Krylov-type methods to compute matrix functions e^{kA} and φ-functions φ₁(kA), φ₂(kA) efficiently and stably.
• Derive exact update formulas for both splitting methods: (75)–(76) for Lie-Trotter and (95), (97), (99) for Strang, incorporating boundary corrections.
• Apply the method to finite-difference spatial discretizations, exploiting block-diagonal structure for computational efficiency.

Experimental results

Research questions

• RQ1 Can order reduction in exponential splitting methods be avoided for nonlinear diffusion-reaction problems with non-homogeneous Dirichlet, Neumann, and Robin boundary conditions?
• RQ2 What modifications to standard Lie-Trotter and Strang splitting are required to preserve full convergence order in time when boundary conditions are non-homogeneous?
• RQ3 How does the error behave after full space-time discretization, and under what conditions is the global error order preserved?
• RQ4 Can the proposed method be practically implemented with efficient matrix function evaluations using Krylov subspace techniques?
• RQ5 Does the theoretical error analysis hold under general spatial discretizations, including finite differences and collocation methods?

Key findings

• The proposed method successfully avoids order reduction in both Lie-Trotter and Strang exponential splitting methods for nonlinear diffusion-reaction problems with non-homogeneous boundary conditions.
• For Lie-Trotter splitting, the global error converges with order 1, and the local error with order 1, as confirmed by numerical experiments with k = 5×10⁻³, 2.5×10⁻³, 1.25×10⁻³.
• For Strang splitting, the global error converges with order 2 when the bound (91) on the spatial discretization is satisfied, as evidenced by convergence orders of 1.9560 and 1.8946 in Table 8.
• The local error for Strang splitting shows second-order convergence (orders 1.8169 and 1.8307), indicating that the method preserves the theoretical order.
• The method is computationally efficient due to the block-tridiagonal structure of spatial operators, enabling fast matrix-vector multiplication via N×N blocks.
• Numerical results in Tables 7 and 8 confirm that the theoretical convergence orders are achieved for both one- and two-dimensional problems using second-order finite differences.