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Convergence rate of the spectral difference method on regular triangular meshes

Pavel Bakhvalov

TL;DR

The paper addresses the convergence rate of the SD-RT method for the 2D transport equation on translationally invariant triangular meshes. It proves that for $p=1$ the method converges with order $p$ when the transport velocity is aligned with a mesh-edge family and with order $p+1$ otherwise; numerical experiments extend this observation to $p=2,3$. The analysis combines a stability study, a mean truncation error analysis on the periodic cell, and a criterion linking truncation-error structure to convergence rate. Numerical results corroborate the theoretical rates and illuminate long-time accuracy under different velocity directions.

Abstract

We consider the spectral difference method based on the p-th order Raviart~-- Thomas space (p=1,2,3) on regular triangular meshes for the scalar transport equation. The solution converges with the order p if the transport velocity is parallel to a family of mesh edges and with the order p+1 otherwise. We prove this fact for p=1 and show it for p=1,2,3 in numerical experiments.

Convergence rate of the spectral difference method on regular triangular meshes

TL;DR

The paper addresses the convergence rate of the SD-RT method for the 2D transport equation on translationally invariant triangular meshes. It proves that for the method converges with order when the transport velocity is aligned with a mesh-edge family and with order otherwise; numerical experiments extend this observation to . The analysis combines a stability study, a mean truncation error analysis on the periodic cell, and a criterion linking truncation-error structure to convergence rate. Numerical results corroborate the theoretical rates and illuminate long-time accuracy under different velocity directions.

Abstract

We consider the spectral difference method based on the p-th order Raviart~-- Thomas space (p=1,2,3) on regular triangular meshes for the scalar transport equation. The solution converges with the order p if the transport velocity is parallel to a family of mesh edges and with the order p+1 otherwise. We prove this fact for p=1 and show it for p=1,2,3 in numerical experiments.
Paper Structure (14 sections, 7 theorems, 36 equations, 7 figures)

This paper contains 14 sections, 7 theorems, 36 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The SD-RT method
  3. The Raviart -- Thomas space
  4. SD-RT on a general triangular mesh
  5. SD-RT(1) on a right-triangular mesh
  6. Criterion of the $(p+1)$-th order convergence
  7. Accuracy analysis of SD-RT(1)
  8. Stability
  9. Properties of $L(0)$
  10. Mean truncation error on the periodic cell
  11. Main result
  12. The long-time simulation accuracy
  13. Numerical results
  14. How to get the coefficients of $\tilde{\Pi}_h$ in Lemma \ref{['th:lemma6']}

Key Result

Proposition 1

Let the scheme eq3 be stable. If for each multiindex $\bm{m}$, $\vert\bm{m}\vert=p+1$, there holds $f^{\bm{m}} \in \mathrm{Im}\,L(0)$, then the optimal order of accuracy of the scheme eq3 is $p+1$. Otherwise the optimal order of accuracy is $p$.

Figures (7)

  • Figure 1: The checkerboard function on a regular triangular mesh
  • Figure 2: Solution points
  • Figure 3: The norm of the solution error at $t=0.1$
  • Figure 4: The norm of the solution error for $p=1$, $\phi=0$
  • Figure 5: The norm of the solution error for $p=1$, $\phi=\pi/8$
  • ...and 2 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 1
  • Theorem 5
  • proof
  • ...and 4 more