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

Yu-Chen Cheng, Christian Klingenberg, Rony Touma

TL;DR

This work addresses numerical solution of hyperbolic balance laws with gravity by introducing a second-order unstaggered central well-balanced scheme that leverages the Deviation method around a hydrostatic state and the KT scheme. By evolving the deviation $\Delta q = q - \tilde{q}$ and preserving the local wave speeds, the method remains Riemann-solver-free while exactly preserving stationary solutions. A fully discrete scheme with Reconstruction, Evolution, and Projection steps is developed, and a semi-discrete form is shown to be essentially TVD for scalar cases, ensuring non-oscillatory behavior. Numerical experiments on 1D Euler equations with gravity—including isothermal equilibria, perturbed and moving equilibria, and shock tubes—validate the well-balanced property, second-order accuracy, and grid-convergence, with minor oscillations only on very coarse grids. The approach offers a practical, robust tool for gravity-driven flows and sets the stage for extensions to higher dimensions.

Abstract

In this paper, we propose a new MUSCL scheme by combining the ideas of the Kurganov and Tadmor scheme and the so-called Deviation method which results in a well-balanced finite volume method for the hyperbolic balance laws, by evolving the difference between the exact solution and a given stationary solution. After that, we derive a semi-discrete scheme from this new scheme and it can be shown to be essentially TVD when applied to a scalar conservation law. In the end, we apply and validate the developed methods by numerical experiments and solve classical problems featuring Euler equations with gravitational source term.

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

TL;DR

This work addresses numerical solution of hyperbolic balance laws with gravity by introducing a second-order unstaggered central well-balanced scheme that leverages the Deviation method around a hydrostatic state and the KT scheme. By evolving the deviation and preserving the local wave speeds, the method remains Riemann-solver-free while exactly preserving stationary solutions. A fully discrete scheme with Reconstruction, Evolution, and Projection steps is developed, and a semi-discrete form is shown to be essentially TVD for scalar cases, ensuring non-oscillatory behavior. Numerical experiments on 1D Euler equations with gravity—including isothermal equilibria, perturbed and moving equilibria, and shock tubes—validate the well-balanced property, second-order accuracy, and grid-convergence, with minor oscillations only on very coarse grids. The approach offers a practical, robust tool for gravity-driven flows and sets the stage for extensions to higher dimensions.

Abstract

In this paper, we propose a new MUSCL scheme by combining the ideas of the Kurganov and Tadmor scheme and the so-called Deviation method which results in a well-balanced finite volume method for the hyperbolic balance laws, by evolving the difference between the exact solution and a given stationary solution. After that, we derive a semi-discrete scheme from this new scheme and it can be shown to be essentially TVD when applied to a scalar conservation law. In the end, we apply and validate the developed methods by numerical experiments and solve classical problems featuring Euler equations with gravitational source term.
Paper Structure (14 sections, 2 theorems, 78 equations, 7 figures, 1 table)

This paper contains 14 sections, 2 theorems, 78 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. A new central well-balanced strategy for in-homogeneous hyperbolic conservation laws
  3. Framework of the Deviation method
  4. Application of the Kurganov-Tadmor scheme
  5. Reconstruction
  6. Evolution
  7. Projection
  8. Semi-discrete scheme
  9. Numerical experiments and validations
  10. Isothermal Equilibrium
  11. Isothermal Equilibrium with Perturbation
  12. Moving Equilibrium
  13. The Shock Tube Problem
  14. Conclusion

Key Result

Lemma 2.1

Consider the balance law 3.1 and a given hydrostatic solution $\tilde{q}$. The deviation quantity $\Delta q$ satisfying the modified balance law 3.7 maintains the same local speed as the original balance law in system 3.1.

Figures (7)

  • Figure 1: Geometry of the NT Scheme: piecewise linear interpolants $L^n(x)$ are evolved on two staggered grids.
  • Figure 2: Geometry of the KT Scheme (displayed by using the deviation $\Delta q$)
  • Figure 3: Results of 1D isothermal equilibrium.
  • Figure 4: Initial perturbation at t=0 compared to the perturbation at the final time t=0.25 and the perturbation from compared solution.
  • Figure 5: Solution of one-dimensional moving equilibrium.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof