An approximate Lax-Wendroff-type procedure for high order accurate schemes for hyperbolic conservation laws

A high order time stepping applied to spatial discretizations provided by the method of lines for hyperbolic conservations laws is presented. This procedure is related to the one proposed in Qiu and Shu (SIAM J Sci Comput 24(6):2185-2198, 2003) for numerically solving hyperbolic conservation laws. Both methods are based on the conversion of time derivatives to spatial derivatives through a Lax-Wendroff-type procedure, also known as Cauchy-Kovalevskaya process. The original approach in Qiu and Shu (2003) uses the exact expressions of the fluxes and their derivatives whereas the new procedure computes suitable finite difference approximations of them ensuring arbitrarily high order accuracy both in space and time as the original technique does, with a much simpler implementation and generically better performance, since only flux evaluations are required and no symbolic computations of flux derivatives are needed.