Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations

TL;DR

This paper develops and analyzes implicit and implicit-explicit ADER and DeC methods for solving ODEs and PDEs, revealing their stability properties and practical applicability for advection-related equations.

Contribution

It introduces new implicit and implicit-explicit ADER and DeC schemes within the DeC framework and provides a detailed stability analysis as Runge-Kutta methods.

Findings

Stability varies from A-stable to bounded depending on method and order.

Methods are stable under CFL-like conditions for advection-diffusion equations.

Explicit stability boundaries are established for different schemes.

Abstract

In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary differential equations (ODEs) and within the context of linear partial differential equations (PDEs). To analyze their stability, we reinterpret these methods as Runge-Kutta schemes and uncover significant variations in stability behavior, ranging from A-stable to bounded stability regions, depending on the chosen order, method, and quadrature nodes. This differentiation contrasts with their explicit counterparts. When applied to advection-diffusion and advection-dispersion equations employing finite difference spatial discretization, the von Neumann stability analysis demonstrates stability under CFL-like conditions. Particularly noteworthy is the stability maintenance observed for the advection-diffusion equation, even under spatial-independent constraints. Furthermore, we establish precise boundaries for relevant coefficients and provide suggestions regarding the suitability of specific schemes for different problem.