[Paper Review] High-order difference and pseudospectral methods for discontinuous problems

C. Markakis, Leor Barack|arXiv (Cornell University)|Jun 18, 2014

Differential Equations and Numerical Methods4 citations

TL;DR

This paper presents a high-order numerical method for solving PDEs with discontinuous solutions by correcting Lagrange interpolation with known jumps in the function and its derivatives. By modifying standard finite-difference or pseudospectral matrices to incorporate these jumps, the method achieves spectral accuracy even in the presence of discontinuities, enabling efficient and accurate solution of evolution PDEs with moving fronts or shocks.

ABSTRACT

High order finite-difference or spectral methods are typically problematic in approximating a function with a jump discontinuity. Some common remedies come with a cost in accuracy near discontinuities, or in computational cost, or in complexity of implementation. However, for certain classes of problems involving piecewise analytic functions, the jump in the function and its derivatives are known or easy to compute. We show that high-order or spectral accuracy can then be recovered by simply adding to the Lagrange interpolation formula a linear combination of the jumps. Discretizations developed for smooth problems are thus easily extended to nonsmooth problems. Furthermore, in the context of one-dimensional finite-difference or pseudospectral discretizations, numerical integration and differentiation amount to matrix multiplication. We construct the matrices for such operations, in the presence of known discontinuities, by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient way to obtain solutions with moving discontinuities to evolution partial differential equations.

Motivation & Objective

• To address the loss of high-order accuracy in finite-difference and pseudospectral methods when solving problems with jump discontinuities.
• To develop a simple, efficient extension of existing smooth-problem discretizations to handle nonsmooth solutions with known jumps.
• To enable high-order accuracy in time-dependent PDEs with moving discontinuities using a method-of-lines framework.
• To construct matrix operators for differentiation and integration that incorporate jump corrections while preserving spectral convergence.

Proposed method

• The method modifies the standard Lagrange interpolation formula by adding a linear combination of known jumps in the function and its derivatives.
• The correction terms are derived analytically from the jump discontinuities, which are assumed to be known or easily computable.
• Finite-difference or pseudospectral matrices are redefined by applying the corrected interpolation formula to construct differentiation and integration operators.
• The resulting matrices are used in a method-of-lines framework to solve time-dependent PDEs with moving discontinuities.
• The approach preserves the structure of standard spectral methods while ensuring high-order accuracy near discontinuities.
• The method avoids the need for adaptive refinement, domain decomposition, or complex interface tracking by embedding jump information directly into the basis functions.

Experimental results

Research questions

• RQ1 Can high-order accuracy be preserved in spectral and finite-difference methods when solving PDEs with known jump discontinuities?
• RQ2 How can standard spectral matrices be modified to incorporate discontinuity information without sacrificing efficiency or accuracy?
• RQ3 What is the impact of jump corrections on the conditioning and stability of pseudospectral discretizations?
• RQ4 Can this approach be seamlessly integrated into a method-of-lines framework for time-evolving PDEs with moving fronts?

Key findings

• The method restores spectral accuracy in problems with jump discontinuities by incorporating known jumps into the interpolation formula.
• The correction terms are derived analytically and added as a linear combination to the standard Lagrange interpolation, preserving high-order convergence.
• Matrix operators for differentiation and integration are constructed using the corrected interpolation, enabling efficient solution of evolution PDEs.
• The approach allows for high-order accuracy in both space and time, even with moving discontinuities, without requiring complex numerical treatments.
• The method is computationally efficient and maintains the simplicity of standard spectral methods, avoiding costly remeshing or adaptive refinement.