Central high resolution schemes capturing discontinuities inside cells via average-interpolating discontinuous radial basis functions: Applications to wave propagation in layered media

Central high-resolution schemes usually consider continuous variation inside cells. Hence, capturing of a possible discontinuity inside a cell would be an open research area. Here, it is tried to capture a (stationary) discontinuity inside a cell by the concept of the discontinuous radial basis functions (RBFs) in the reconstruction stage of central high-resolution schemes. As the formulations of central high-resolution schemes are in the framework of the Godunov method, firstly the concept of the point-wise interpolating discontinuous RBFs is extended to average interpolating discontinuous RBFs. At the next stage, by using this special reconstruction (by considering a discontinuity inside a cell), the formulation of the fully-discrete form is derived. Corresponding semi-discrete form is then obtained in the limiting state, as the time step, Δt approaches zero. In the reconstruction stage, for a cell with a possible discontinuity inside the cell, discontinuous RBFs with C0 continuity feature are used and for other cells (without inner discontinuities), smooth RBFs with C2 continuity property are utilized. Here, the Wendland family of 1-D RBFs is considered with different continuity properties. This central formulation would be useful for problems with stationary discontinuities. Finally, different 1-D and 2-D benchmarks with stationary discontinuities are examined including stress wave propagation problems in layered media. The benchmarks and examples include conservation laws with space-dependent flux functions.