Approximate Lax-Wendroff discontinuous Galerkin methods for hyperbolic conservation laws

The Lax-Wendroff time discretization is an alternative method to the popular total variation diminishing Runge-Kutta time discretization of discontinuous Galerkin schemes for the numerical solution of hyperbolic conservation laws. The resulting fully discrete schemes are known as LWDG and RKDG methods, respectively. Although LWDG methods are in general more compact and efficient than RKDG methods of comparable order of accuracy, the formulation of LWDG methods involves the successive computation of exact flux derivatives. This procedure allows one to construct schemes of arbitrary formal order of accuracy in space and time. A new approximation procedure avoids the computation of exact flux derivatives. The resulting approximate LWDG schemes, addressed as ALWDG schemes, are easier to implement than their original LWDG versions. In particular, the formulation of the time discretization of the ALWDG approach does not depend on the flux being used. Numerical results for the scalar and system cases in one and two space dimensions indicate that ALWDG methods are more efficient in terms of error reduction per CPU time than LWDG methods of the same order of accuracy. Moreover, increasing the order of accuracy leads to substantial reductions of numerical error and gains in efficiency for solutions that vary smoothly.