Cardinal-type approximations for conservation laws of mixed type

Nonlinear partial differential equations appear in many branches of physics, engineering and applied mathematics. This paper provides a technical description of the application of collocation interpolation methods based on Sinc functions to conservation laws of mixed hyperbolic-elliptic type, with Riemann type conditions. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The nonlinear terms are easily handled with the help of Hadamard matrix multiplications. The error in the solution is shown to converge to the exact solution at an exponential rate. The convergence proof of the solution for the discrete system is given using fixed-point iteration. The scheme is numerically tested on the Van der Waals equation in fluid dynamics. Easy and economical implementation is the strength of this method.