Abstract: Near-equilibrium solutions are the main interest in the modelling, analysis and numerical approximation of hyperbolic balance laws. Numerical approximations have to discretize both conservative flux differences, usually via Riemann solvers, and non-conservative source term, via quadrature. Both the conservative and the nonconservative forces may be very large, while the perturbations from equilibria are usually much smaller. Therefore, it is not enough to have small truncation errors compared to the size of each of these forces. The truncation error must also be small with respect to the perturbation itself. This challenge has led to many so-called well-balanced schemes, especially for the shallow water equations with topography and friction, and for the gravitational Euler equations, usually in one space dimension.

For each case, a special study of equilibrium variables led to particular reconstructions and quadrature rules for the source term. Recently, in a bold attempt, Chertock, Herty & Ozcan reformulated one-dimensional systems of balance laws as conservation laws by integrating the source term and adding it to the flux difference. Reconstructing the combined flux (called equilibrium flux) and discretizing the new conservation law using conservative flux differences leads to automatic well-balancing for any system and any equilibrium state. This is surprising, because we are used to think of conservation laws and balance laws as different types of partial differential equations.

In the first part of this talk we will reconcile this apparent paradox, and argue that the new equilibrium flux system is only a piecewise local conservation law, and that its numerical fluxes should be one-sided and hence not conservative.

In the second part of the talk, I will discuss work in progress which generalizes the Chertock / Herty / Ozcan approach to multiple space dimensions. Instead of integrating the source term, we solve local Poisson problems with Neumann boundary conditions for each cell. If this approach will turn out to be successful, we may be able to develop well-balanced, high order discontinuous Galerkin schemes in multi-D, and for general systems of balance laws.

Our argument is based on a recent, semi-discrete in time, piecewise smooth in space, local weak formulation of multi-dimensional balance laws.