This talk is dedicated to the presentation of several recent papers, all dealing with the development of fully well-balanced (FWB) methods, i.e. numerical methods which exactly (or approximately) preserve the steady solutions of a system of hyperbolic balance laws. In addition, the schemes we describe share another property: they do not require the costly inversion of nonlinear systems.
Namely, we will present results from https://hal.science/hal-03271103/document, https://hal.science/hal-04083181/document and https://hal.science/hal-04246991/document:
1/ A high-order FWB scheme obtained by introducing a straightforward correction method, applicable to schemes of order 2 or higher, such as MUSCL-type schemes. This correction ensures exact preservation of steady solutions without the need to invert the underlying nonlinear system. This technique ends up being a way of making any first-order scheme exactly well-balanced, but it relies on a first-order FWB scheme to fall back to.
2/ To that end, we also present an extension of the well-known hydrostatic reconstruction to preserve steady solutions of the shallow water system with nonzero velocity, without the need for specific numerical fluxes, and without having to solve a nonlinear system.
3/ Finally, relaxing the constraint on "exact" preservation of the steady solution, we design new discontinuous Galerkin (DG) basis functions able to either exactly or approximately preserve steady solutions. The DG basis is enriched with a prior computed by a Physics-Informed Neural Network (PINN), maintaining the same convergence order but improving the error constant.