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A Globally Conservative Compact Framework for Conservation Laws: Fourth-Order Schemes with Enhanced Resolution and Stability

Weifeng Hou, Zhangpeng Sun, Wenqi Yao, Liupeng Wang

Abstract

The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.

A Globally Conservative Compact Framework for Conservation Laws: Fourth-Order Schemes with Enhanced Resolution and Stability

Abstract

The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.

Paper Structure

This paper contains 17 sections, 2 theorems, 79 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Globally conservative compact finite difference framework
  3. Preliminary review
  4. A globally conservative framework
  5. Globally fourth-order conservative schemes
  6. Analysis and optimization
  7. Fourier analysis
  8. Optimization
  9. Numerical examples
  10. One-dimensional linear advection
  11. Two-dimensional advection: non-homogeneous coefficients
  12. Two-dimensional inviscid vortex convection in a supersonic flow
  13. Conclusion
  14. Appendix
  15. The necessary condition for the construction of schemes P1, P2 and P3
  16. ...and 2 more sections

Key Result

Theorem 1

Denote $T_I^i[f]$, $T_L^i[f]$ and $T_R^i[f]$ as the truncation errors of Eq. matrix_form_derivative on $x_i\in$$\mathcal{G}_I$, $\mathcal{G}_b^l$ and $\mathcal{G}_b^r$, respectively. Let (C1)-(C3) be fulfilled. Then, Eq. numerical_integration with the integration weight vector $W$ satisfy Eq. wp_relation_w posesses $2k-1$-th order algebraic precision.

Figures (9)

  • Figure 1: Distribution of eigenvalues of $Q_{P1}$, $Q_{P2}$ and $Q_{P3}$.
  • Figure 2: The real parts of pseudo-wavenumber at nodes $i=0,1,2$
  • Figure 3: The imaginary parts of pseudo-wavenumber at nodes $i=0,1,2$
  • Figure 4: Maximum error in $u$ over $0\leq t\leq1000$, The expected order of each scheme is drawn as a solid black line.
  • Figure 5: $L_\infty$ norm of error over $0\leq t\leq1000$, for indicated $N$ and constant timestep of $\Delta t=0.001$ for the varying coefficient scalar wave equation test.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof