The ADER Approach for Approximating Hyperbolic Equations to Very High Accuracy

Sixty years ago, Godunov introduced his method for solving the Euler equations of gas dynamics, thus creating the Godunov's school of thought for the numerical approximation of hyperbolic equations. The building block of the original first-order Godunov upwind method is the solution of the conventional piecewise constant data Riemann problem. The ADER methodology is a high-order, non-linear fully discrete one-step extension of Godunov's method. The building block of an ADER scheme of order m + 1 in space and time is the generalized Riemann problem GRP(m), in which source terms are admitted and the initial data is represented by polynomials of arbitrary degree m, or other functions. There are by now several methods available to solve theupper GRP(m). The ADER fully discrete methodology operates in both the finite volume and DG frameworks, containing all orders of accuracy. Here we review some key aspects of ADER and conclude with a practical example that highlights the key point of very high-order methods: for small errors they are orders-of-magnitude more efficient than low order methods.