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A Well-Balanced Method for an Unstaggered Central Scheme, the two-space Dimensional Case

Yu-Chen Cheng, Christian Klingenberg, Rony Touma

TL;DR

The work addresses the challenge of solving 2D hyperbolic balance laws with gravity while preserving hydrostatic equilibria. It blends the Deviation method with the two-dimensional Kurganov–Tadmor (KT) scheme to form a well-balanced, central, Riemann-solver-free framework that avoids oversampling and remains non-oscillatory. The authors develop both fully discrete and semi-discrete formulations, prove a maximum-principle property for the semi-discrete scheme, and validate the approach on the 2D Euler equations with gravity through isothermal equilibria, perturbations, moving equilibria, and shock-tube tests, achieving second-order accuracy. The method shows robust performance in preserving equilibria and resolving shocks, with potential applicability to a broader class of systems with source terms.

Abstract

We develop a second-order accurate central scheme for the two-dimensional hyperbolic system of in-homogeneous conservation laws. The main idea behind the scheme is that we combine the well-balanced deviation method with the Kurganov-Tadmor (KT) scheme. The approach satisfies the well-balanced property and retains the advantages of KT scheme: Riemann-solver-free and the avoidance of oversampling on the regions between Riemann-fans. The scheme is implemented and applied to a number of numerical experiments for the Euler equations with gravitational source term and the results are non-oscillatory. Based on the same idea, we construct a semi-discrete scheme where we combine the above two methods and illustrate the maximum principle.

A Well-Balanced Method for an Unstaggered Central Scheme, the two-space Dimensional Case

TL;DR

The work addresses the challenge of solving 2D hyperbolic balance laws with gravity while preserving hydrostatic equilibria. It blends the Deviation method with the two-dimensional Kurganov–Tadmor (KT) scheme to form a well-balanced, central, Riemann-solver-free framework that avoids oversampling and remains non-oscillatory. The authors develop both fully discrete and semi-discrete formulations, prove a maximum-principle property for the semi-discrete scheme, and validate the approach on the 2D Euler equations with gravity through isothermal equilibria, perturbations, moving equilibria, and shock-tube tests, achieving second-order accuracy. The method shows robust performance in preserving equilibria and resolving shocks, with potential applicability to a broader class of systems with source terms.

Abstract

We develop a second-order accurate central scheme for the two-dimensional hyperbolic system of in-homogeneous conservation laws. The main idea behind the scheme is that we combine the well-balanced deviation method with the Kurganov-Tadmor (KT) scheme. The approach satisfies the well-balanced property and retains the advantages of KT scheme: Riemann-solver-free and the avoidance of oversampling on the regions between Riemann-fans. The scheme is implemented and applied to a number of numerical experiments for the Euler equations with gravitational source term and the results are non-oscillatory. Based on the same idea, we construct a semi-discrete scheme where we combine the above two methods and illustrate the maximum principle.
Paper Structure (24 sections, 2 theorems, 98 equations, 11 figures, 1 table)

This paper contains 24 sections, 2 theorems, 98 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. A new two-dimensional scheme for in-homogeneous conservation laws
  3. Framework of the two-dimensional Deviation method
  4. Application of the two-dimensional Kurganov-Tadmor-type scheme
  5. Reconstruction
  6. Evolution
  7. Projection
  8. 2D Semi-discrete scheme
  9. Construction of the fully-discrete scheme
  10. Reconstruction
  11. Evolution
  12. Projection
  13. New semi-discrete scheme
  14. Maximum Principle
  15. Numerical experiments and validations
  16. ...and 9 more sections

Key Result

Lemma 2.1

Consider the balance law 4.1 and a given hydrostatic solution $\tilde{q}$. The deviation quantity $\Delta q$ satisfying the modified balance law 4.5 maintains the same local speeds as those in the original balance system 4.1.

Figures (11)

  • Figure 1: Computational cells are split into smooth and unsmooth non-uniform quadrilateral subdomains by the maximal local wave speeds.
  • Figure 2: Nine subdomains $C^I_{j,k}$ of the original control cell $C_{j,k}$ and their centers of mass.
  • Figure 3: A general quadrilateral with normal $\eta$ split into two cases.
  • Figure 4: Computational cells are split into smooth and unsmooth non-uniform rectangular subdomains by the maximal local wave speeds
  • Figure 5: Results of 2-d isothermal equilibrium: the top two figures are the results from our new scheme; the two below are the exact solutions.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 3.1: Maximum principle
  • proof