Solving hyperbolic conservation laws using multiquadric quasi-interpolation

In this article, we apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi-interpolation corresponding to periodic and inflow-outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the differential equation and a low-order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one-dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is O(tau), where tau is the temporal step. We can improve the accuracy by using the high-order quasi-interpolation. Moreover the methods can be generalized to the other equations. (C) 2005 Wiley Periodicals, Inc.