Dynamical knot and shape parameter setting for simulating shock wave by using multi-quadric quasi-interpolation

In order to solve non-linear wave equation numerically, three features are demonstrated in this paper. First we move the knots dynamically to keep the data denser near the shocks than elsewhere. Second we set the shape parameters of the multi-quadric dynamically to get an efficient simulation of the shock wave. Finally we use quasi-interpolation served as viscosity step by step respected to the discrete time level. Theoretical discussion for the shape preserving and the variation diminishing property (the base point of the stability of the scheme) is showed in the paper too. Numerical example shows that the method works very well.