ADER High-Order Methods

This chapter is concerned with advanced, numerical methods for evolutionary partial differential equations, following the ADER framework. The methodology is an unlimited-order, non-linear fully discrete one-step extension of Godunov’s method that operates in the finite volume and discontinuous Galerkin finite element frameworks. The approach is applicable to multidimensional systems in conservative and non-conservative forms, on structured and unstructured meshes. The building block of ADER is the generalised Riemann problem \(GRP_{m}\) , admitting stiff and non-stiff source terms, with initial data consisting of reconstruction polynomials of arbitrary degree m. The resulting ADER schemes are of order \(m+1\) in both space and time. Here we review some key aspects of the methodology and give appropriate references. The presentation is general, so as to include any system of hyperbolic balance laws and extensions, but we also identify specific applications to the shallow water equations of interest in this book. The chapter is concluded with two examples that highlight the key point of very high-order methods; that is, for small errors high-order ADER methods are orders-of-magnitude more efficient than low-order methods.