Well-Balanced Schemes and Path-Conservative Numerical Methods

Abstract In this chapter we describe a general methodology for developing high-order well-balanced schemes for hyperbolic system with nonconservative products and/or source terms. We briefly recall the Dal Maso–LeFloch–Murat theory to define weak solutions of nonconservative systems and how it has been used to establish the notion of path-conservative schemes. We show that, under this framework, it is possible to extend to the nonconservative case many well-known numerical schemes that are commonly used for system of conservation laws. Moreover, their extension to high order can be done as well. Next the well-balanced property of the proposed methods is analyzed with an illustrative 1d example. Finally, we point out the difficulties related to the right definition of weak solution and the design of numerical schemes converging to them.

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