A well-balanced adaptive Haar wavelet finite volume scheme for 1D free surface water flows

Abstract

Abstract This paper studies a Haar-wavelet based finite volume method in terms of its capability of preserving a well-balanced property compared to the classical finite volume method, as well as with its application on the real case of the shallow water flow which has irregular bed elevation. The well-balanced property is considered as an important benchmark for validation of the numerical scheme, particularly when the system of the governing equations is inhomogeneous (i.e. the source term is not zero). Haar wavelet finite volume scheme is obtained via merging the theory of wavelets with the formulation of finite volume method. The Roe approximate Riemann solver is adopted to evaluate the numerical fluxes at cell boundaries. The source term is represented by the frictionless topography in the test case. Furthermore, the numerical results are compared to the classical finite volume scheme and the real case and it is in a good agreement with the exact and reference finite volume scheme in which the adaptive scheme preserved the equilibrium state of the system. The adaptive scheme inherits the features of the classical finite scheme but with less computational effort in which the adaptive scheme needs a maximum of 211 cells (8.8% of the finest uniform mesh refinement) to get the convergence of the scheme. While in the partially wet sub-case, it needs 223 cells (9.3% of the finest uniform mesh refinement). The numerical scheme was preserving the root mean square error of the computational results within the range of the machine precision.