[Paper Review] High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

Jonas P. Berberich, Praveen Chandrashekar+1 more|arXiv (Cornell University)|Mar 12, 2019

Computational Fluid Dynamics and Aerodynamics4 citations

TL;DR

This paper presents a general high-order well-balanced finite volume framework for multi-dimensional hyperbolic balance laws that exactly maintains any known stationary or time-dependent solution—analytical or discrete—by evolving deviations from the target solution. The method preserves high-order accuracy, requires minimal code changes, and incurs only a modest 5–30% increase in computational cost, even for high-order schemes.

ABSTRACT

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase well-balancing in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.

Motivation & Objective

• To develop a general framework for constructing high-order well-balanced finite volume methods applicable to any system of hyperbolic balance laws in multiple spatial dimensions.
• To enable exact preservation of arbitrary known solutions—static, stationary, or time-dependent—without restricting to specific equations of state or hydrostatic assumptions.
• To maintain high-order accuracy while modifying only the source term discretization and reconstruction strategy, ensuring compatibility with existing finite volume solvers.
• To demonstrate robustness and efficiency on complex systems such as compressible Euler equations with gravity and ideal MHD, including non-trivial stationary states with non-zero velocity.
• To quantify the computational cost increase of the well-balancing modification across different orders of accuracy and solution types.

Proposed method

• The method reformulates the finite volume scheme to evolve the deviation from a known target solution rather than the solution itself, ensuring exactness on the target.
• It uses hydrostatic reconstruction principles adapted to arbitrary target solutions, with reconstruction based on the known solution's structure.
• The source term is modified via a well-balanced quadrature strategy that exactly balances flux and source terms at the discrete level.
• The framework is compatible with any consistent reconstruction, numerical flux, quadrature rule, and time integrator, enabling arbitrary high-order accuracy.
• The approach is implemented on structured Cartesian and polar grids, supporting both static and time-evolving target solutions.
• For discrete target solutions (e.g., from numerical simulations), the method uses precomputed data arrays to maintain accuracy and efficiency.

Experimental results

Research questions

• RQ1 Can a general finite volume framework be designed to exactly preserve any known solution of a hyperbolic balance law, regardless of the system or solution type?
• RQ2 How can high-order accuracy be maintained in well-balanced schemes when the solution is not a simple static state?
• RQ3 What is the computational overhead of enforcing well-balancing for time-dependent or complex stationary solutions?
• RQ4 Can the method be applied to systems with complex equations of state or non-trivial velocity fields, such as differentially rotating stars?
• RQ5 How does the computational cost of the well-balancing modification scale with increasing order of accuracy?

Key findings

• The method exactly maintains static and stationary solutions of the compressible Euler equations with gravity and ideal MHD, even with complex equations of state or non-zero velocity fields.
• For a one-dimensional static solution, the computational cost increased by only about 20% for first- and second-order methods, and by just 5% for seventh-order methods.
• On a polar grid with a Keplerian disk (stationary, time-independent), the cost increase was less than 20% for first-order and less than 10% for second-order well-balanced schemes.
• For a time-dependent solution (Euler wave with perturbation), the wall-clock time ratio reached up to 30% for first-order, but only about 15% for third-order methods.
• The method maintains high-order accuracy in all numerical tests, with observed convergence rates matching theoretical expectations.
• The framework is easily implementable in existing finite volume codes with minimal modifications, primarily involving source term and reconstruction adjustments.