Approximate well-balanced WENO finite difference schemes using a global-flux quadrature method with multi-step ODE integrator weights
Maria Kazolea, Carlos Parés Madroñal, Mario Ricchiuto
TL;DR
This work develops high-order well-balanced finite-difference schemes for one-dimensional hyperbolic balance laws by embedding the steady-state consistency of high-order ODE integrators directly into a global-flux WENO framework. The source term is integrated on the grid with weights from multi-step ODE solvers (e.g., Adams-Bashforth/Moulton up to order 8), ensuring discrete stationary states align with the underlying ODE solutions without explicit local Cauchy solves. Discrete steady states are shown to converge with the order of the ODE method when $F(U)$ is invertible, and the approach is extended to handle singular sources and exact water-at-rest preservation via two practical strategies. Numerical tests on Burgers’ and shallow-water equations demonstrate optimal time-dependent convergence governed by the minimum of spatial and ODE orders, drastic reductions in steady-state error, and robust handling of discontinuities and moving equilibria, including those with friction and complex bathymetry.
Abstract
In this work, high-order discrete well-balanced methods for one-dimensional hyperbolic systems of balance laws are proposed. We aim to construct a method whose discrete steady states correspond to solutions of arbitrary high-order ODE integrators. However, this property is embedded directly into the scheme, eliminating the need to apply the ODE integrator explicitly to solve the local Cauchy problem. To achieve this, we employ a WENO finite difference framework and apply WENO reconstruction to a global flux assembled nodewise as the sum of the physical flux and a source primitive. The novel idea is to compute the source primitive using high-order multi-step ODE methods applied on the finite difference grid. This approach provides a locally well-balanced splitting of the source integral, with weights derived from the ODE integrator. By construction, the discrete solutions of the proposed schemes align with those of the underlying ODE integrator. The proposed methods employ WENO flux reconstructions of varying orders, combined with multi-step ODE methods of up to order 8, achieving steady-state accuracy determined solely by the ODE method's consistency. Numerical experiments using scalar balance laws and shallow water equations confirm that the methods achieve optimal convergence for time-dependent solutions and significant error reduction for steady-state solutions.