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

TL;DR

The paper develops implicit and implicit–explicit DeC and ADER time-marching methods within the DeC framework and analyzes their stability by recasting them as Runge–Kutta schemes. It demonstrates that stability behavior varies significantly with order, node choice, and IMEX splitting, spanning $A$-stable to bounded regions, and extends the analysis to PDEs using von Neumann stability with CFL-type constraints. Key findings show that ImADER with Gauss–Lobatto nodes attains true $A$-stability via Padé approximations, while equispaced nodes can exhibit high-order instabilities; IMEX schemes exhibit region-dependent stability characterized by several stability notions ($\mathcal{D}_0$, $\mathcal{D}_1$, etc.). Numerical tests on ODEs and PDEs validate convergence orders and stability boundaries, confirming the practical advantage of implicit and IMEX formulations for stiff and advection–diffusion–dispersion problems. The results offer concrete guidelines for selecting schemes and nodes based on problem stiffness, desired accuracy, and CFL-like constraints relevant to finite-difference PDE discretizations.

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.

Figures (18)

• Figure 1: Implicit DeC (left), sDeC (center) and ADER (right) with equispaced (top) and Gauss-Lobatto (bottom) nodes for orders 2 to 13
• Figure 2: Implicit sDeC for orders 14 to 20
• Figure 3: Zoomed stability region of various implicit schemes
• Figure 4: Minion's stability region for IMEX DeC (left), sDeC (center) and ADER (right) with equispaced (top) and GLB (bottom) nodes
• Figure 5: $\mathcal{D}_0$ Stability regions for IMEX ADER. The smaller stability region displays order 2, while the larger one displays order 3 and 4 (right)

Theorems & Definitions (25)

• remark 2.1: Variations of DeC
• remark 2.2: Number of stages
• remark 2.3: Runge-Kutta for sDeC
• remark 3.1: ADER as DeC and order of accuracy of \ref{['eq:ADER_fix_point']}
• Proposition 1: ADER right-hand side
• theorem 3.2: wanner1996solving
• theorem 3.3: wanner1996solving
• theorem 3.4: wanner1996solving
• proposition 3.5: Zero determinant of $\underline{\underline{A}}\, -\underline{1}b^T$
• proof