Finite element methods for nonlinear advection. (English) Zbl 0573.76005

The Taylor-Galerkin method for the temporal and spatial discretization of mixed initial boundary value problems is applied to derive numerical schemes for solving a single scalar conservation-law equation. The discretization in time is performed before the spatial approximation by introducing second-order and third-order accurate generalizations of the standard two-level Euler scheme with the help of Taylor series expansions in the time step. The equations in weak form are then spatially discretized by means of the conventional Galerkin finite element method to obtain a new class of parameter-free, one-step, linearly implicit schemes for the solution of nonlinear hyperbolic problems. Numerical results for the so-called inviscid Burgers’ equation in one and two dimensions are presented to illustrate the properties of the proposed Taylor-Galerkin schemes.


76M99 Basic methods in fluid mechanics


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