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Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge--Kutta discontinuous Galerkin method

Zheng Sun

Abstract

The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the $P^k$ approximation in the final stage, even if we drop the $k$th-order polynomial modes and use the $P^{k-1}$ approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed in arXiv:2402.15150. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability and accuracy of the original method.

Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge--Kutta discontinuous Galerkin method

Abstract

The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the approximation in the final stage, even if we drop the th-order polynomial modes and use the approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed in arXiv:2402.15150. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability and accuracy of the original method.
Paper Structure (49 sections, 30 theorems, 161 equations, 2 figures, 8 tables)

This paper contains 49 sections, 30 theorems, 161 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. RKDG method and its variant.
  3. A truncation error argument for optimal accuracy.
  4. Limitations of the existing analysis.
  5. Literature review.
  6. Challenges, main ideas, and main results.
  7. Significance and novelty.
  8. Organization of the paper.
  9. Notations and preliminaries
  10. DG spatial discretization
  11. 1D case
  12. 2D case
  13. Notations for both 1D and 2D cases.
  14. RKDG and sdA-RKDG schemes
  15. Stability analysis
  16. ...and 34 more sections

Key Result

Proposition 2.1

In both 1D and 2D,

Figures (2)

  • Figure 5.1: Stability verification for 1D RKDG and sdA-RKDG schemes. The $x$ axis is the CFL number. The $y$ axis is the value of $\delta$ defined in \ref{['eq:delta']}.
  • Figure 5.2: Stability verification for 2D RKDG and sdA-RKDG schemes. The $x$ axis is the CFL number. The $y$ axis is value of $\delta$ defined in \ref{['eq:delta']}.

Theorems & Definitions (55)

  • Proposition 2.1: Negative semi-definiteness of the DG operator
  • Remark 2.2: Efficiency
  • Remark 2.3: RK method in Butcher form
  • Theorem 3.1: Stability of sdA-RK2DG1
  • Proposition 3.2: Energy identity of RK2
  • Proposition 3.3: Bounding high-order DG derivatives hy jumps
  • Proposition 3.4: Monotonicity stability of RK2DG1
  • Lemma 3.5: Bounding highest-order DG modes by jumps
  • Lemma 3.6
  • proof
  • ...and 45 more