The Correct Use of the Lax – Friedrichs Method ∗

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of the initial data cannot increase by the application of the scheme. The usefulness of the recipe is confirmed by numerical tests. Mathematics Subject Classification. 35L65, 65M06, 65M12. Received: October 8, 2003. Introduction We are concerned with the operator corresponding to the explicit Lax–Friedrichs method for the approximation of scalar conservation laws. The term operator denotes the abstract summary of the whole procedure under consideration, i.e. in particular including the discretizations of the initial condition, of the computational domain and of the boundary conditions. In the following, this operator is referred to as the LF-Operator. The well-known Lax–Friedrichs scheme proposed by Lax in 1954 [4] we investigate reads as U j = 1 2 ( U j+1 + U n j−1 ) − 1 2 λ [ f ( U j+1 ) − f (Un j−1) ] , (1) where j denotes the spatial index at j∆x, k ∈ {n, n + 1} denotes the temporal level k∆t, λ := ∆t/∆x is the abbreviation of the ratio of mesh widths, and where we employ U as a notation for discrete data. For simplicity, we generally assume that the grid is uniform and that λ is a constant. We also assume the validity of a CFL condition in a strict sense, i.e. we use that for all the values φ in the interval of given data I holds λf ′ (φ) < 1. (2)