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Upwind summation-by-parts finite differences: error estimates and WENO methodology

Yan Jiang, Siyang Wang

TL;DR

The paper analyzes high-order upwind SBP-SAT finite difference schemes for hyperbolic PDEs, using normal-mode analysis to show that the SAT penalty parameter $\tau$ can influence convergence rates (with $\tau \le -\tfrac12$ and a special $\tau=-1$ yielding a $1.5$-order gain). It extends the framework to ESWENO by stabilizing SBP-WENO discretizations, enabling energy-stable, high-order, nonoscillatory solutions for discontinuous data. The authors provide rigorous error estimates for both advection and hyperbolic systems and corroborate them with numerical experiments across smooth and nonsmooth regimes. These results guide practical choices of $\tau$ and demonstrate how ESWENO integrates nonoscillatory capabilities with the SBP-SAT energy-stability framework for robust wave-propagation simulations.

Abstract

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous-approximation-term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

Upwind summation-by-parts finite differences: error estimates and WENO methodology

TL;DR

The paper analyzes high-order upwind SBP-SAT finite difference schemes for hyperbolic PDEs, using normal-mode analysis to show that the SAT penalty parameter can influence convergence rates (with and a special yielding a -order gain). It extends the framework to ESWENO by stabilizing SBP-WENO discretizations, enabling energy-stable, high-order, nonoscillatory solutions for discontinuous data. The authors provide rigorous error estimates for both advection and hyperbolic systems and corroborate them with numerical experiments across smooth and nonsmooth regimes. These results guide practical choices of and demonstrate how ESWENO integrates nonoscillatory capabilities with the SBP-SAT energy-stability framework for robust wave-propagation simulations.

Abstract

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous-approximation-term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.
Paper Structure (15 sections, 6 theorems, 83 equations, 4 figures)

This paper contains 15 sections, 6 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spatial approximation
  3. SBP property
  4. An SBP-SAT discretization
  5. Accuracy analysis for the linear schemes
  6. Error estimate for the linear scheme for the advection equation
  7. Error estimate for the linear scheme for a first order hyperbolic system
  8. An upwind SBP-SAT discretization with the WENO property
  9. Interior order $p=3$
  10. Interior order $p=4$
  11. Stabilization
  12. Numerical experiments
  13. Convergence rate for the linear schemes
  14. SBP-WENO
  15. Conclusion

Key Result

Lemma 1

The semidiscretization semi_adv with $g(t)\equiv 0$ satisfies the energy estimate $\frac{d}{dt}\|\mathbf{u}\|_H^2\leq 0$ if the penalty parameter $\tau\leq -1/2$.

Figures (4)

  • Figure 1: $L^2$ error for solving the advection equation \ref{['eqn_adv']} with different values of $p$ and $\tau$.
  • Figure 2: $L^2$ error solving the hyperbolic system \ref{['hs']} with different values of $p$ and $\tau_1$ for the hyperbolic system.
  • Figure 3: $L^2$ error of the SBP-WENO scheme for $p=3$ (left) and 4 (right) with different values of $\tau$ for the advection equation.
  • Figure 4: Solutions computed with 401 grid points. Left: SBP-WENO scheme with $p=3$ and 4. Right: SBP scheme without WENO.

Theorems & Definitions (13)

  • Definition 1: traditional SBP
  • Definition 2: upwind SBP
  • Lemma 1
  • Proof 1
  • Theorem 1
  • Proof 2
  • Lemma 2: Lemma 2 in Wang2017
  • Theorem 2
  • Proof 3
  • Lemma 3
  • ...and 3 more