Approximate Lax-Wendroff discontinuous Galerkin methods for hyperbolic conservation laws

The Lax-Wendro↵ 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 e cient than RKDG methods of comparable order of accuracy, the formulation of LWDG methods involves the successive computation of exact flux derivatives. This procedure allows 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 ALDWG schemes, are easier to implement than their original LWDG versions. Numerical results for the scalar and system cases in one and two space dimensions indicate that ALWDG methods are more e cient in terms of error reduction per CPU time than LWDG method of the same order of accuracy. Moreover, increasing the order of accuracy leads to substantial reductions of numerical error and gains in e ciency for solutions that vary smoothly.