Algorithms of very high space-time orders of accuracy for hyperbolic equations in the semidiscrete WENO-DeC framework

TL;DR

This work develops and analyzes a truly arbitrary high-order finite-volume framework for hyperbolic PDEs by fusing WENO spatial reconstruction with Deferred Correction time integration (WENO–DeC). It demonstrates that pairing high-order spatial discretization with matching high-order time integrators yields substantial accuracy and efficiency gains across linear and nonlinear tests, while low-order time integration degrades performance even when spatial accuracy is high. Through extensive 1D tests on the linear advection equation and Euler equations, the study clarifies the roles of reconstruction in characteristic variables and the choice of flux (exact Riemann solver vs. Rusanov) in controlling oscillations and resolution of discontinuities. The results confirm the potential of WENO–DeC as a baseline for real-world hyperbolic problems, while also highlighting areas (robustness, limiting, multidimensional extensions) for future work.

Abstract

In this work, we provide a deep investigation of a family of arbitrary high order numerical methods for hyperbolic partial differential equations (PDEs), with particular emphasis on very high order versions, i.e., with order higher than 5. More in detail, within the context of a generic Finite Volume (FV) semidiscretization, we consider Weighted Essentially Non--Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The goal of this paper is twofold. On the one hand, we want to demonstrate the possibility of utilizing very high order schemes in concrete situations and highlight the related advantages. On the other one, we want to debunk the myth according to which, in the context of numerical resolution of hyperbolic PDEs with very high order spatial discretizations, the adoption of lower order time discretizations, e.g., strong stability preserving (SSP) or linearly strong stability preserving ($\ell SSP$) Runge--Kutta (RK) schemes, does not affect the overall accuracy of the resulting approach and consequently its computational efficiency. Numerical results are reported for the linear advection equation (LAE) and for the Euler equations of fluid dynamics, showing the advantages and the critical aspects of the adoption of very high order numerical methods. Overall, the results indicate the potential for their use in real--life applications, offering advantages in terms of efficiency, such as requiring shorter computational times to achieve a prescribed error, even in problems involving discontinuities. Furthermore, the results confirm order degradation and efficiency loss when coupling very high order space discretizations with lower order SSPRK time discretizations.

Figures (18)

• Figure 1: LAE, Test 1: Results obtained with WENO--DeC
• Figure 2: LAE, Test 1: Results obtained with WENO--SSPRK3 and WENO--SSPRK4 on top and efficiency comparison with WENO--DeC on the bottom
• Figure 3: LAE, Test 1: Expected computational times in seconds to reach an accuracy level equal to $10^{-16}$ in the $L^1$-, $L^2$-, and $L^{\infty}$-norms.
• Figure 4: LAE, Test 1: Results obtained with WENO--mSSPRK3 and WENO--mSSPRK4 on top and efficiency comparison with WENO--DeC on the bottom
• Figure 5: LAE, Test 2: Results obtained with WENO--DeC with final time $T_f:=2000$ and several levels of mesh refinement

Theorems & Definitions (5)

• Remark 1: Reconstruction of characteristic variables
• Remark 2: On some critical aspects of the computation of the linear weights
• Remark 3: On the accuracy of the coefficients of SSPRK(5,4)
• Remark 4: On the asymptotic character of convergence
• Remark 5: On smaller final times