We show that any conservative scheme for solving scalar conservation laws in two space dimensions, which is total variation diminishing, is at most first-order accurate.

Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the...

We propose a discretization scheme for a numerical solution of elliptic PDE’s, based on local representation of functions, by their Taylor polynomials (jets). This scheme utilizes jet calculus to provide a very high order of accuracy for a relatively small number of unknowns involved.

Residual distribution schemes are discussed in the context of higher order spatial discretization for hyperbolic conservation laws. The discrete solution is globally continuous, defined by a Finite Element space based on triangular Lagrangian Pk elements. A natural subtriangulation of these elements allows the reuse of simple distribution formula previously developed for linear P1 triangles. Th...

Abstract. We are concerned with the detection of edges – the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our...

Here we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on gener...

Invariant numerical schemes possess properties that may overcome the numerical properties of most of classical schemes. When they are constructed with moving frames, invariant schemes can present more stability and accuracy. The cornerstone is to select relevant moving frames. We present a new algorithmic process to do this. The construction of invariant schemes consists in parametrizing the sc...

and Applied Analysis 3 on the whole space H, is a bounded operator. Here, I is the identity operator. The following operators D ( I τA τA 2 2 ) , G ( I − τ 2A 2 ) , P ( I τ 2 A ) , R I τB −1, Tτ ( I B−1A ( I τA τ 2 P−2 ) K ( I − R2N−1 ) GKP−2R2N−1 −GKP−2 2I τB R [ n ∑ i 1 αi ( I ( λi − [ λi τ ] τ ) A ) D − λi/τ u0 ])−1