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Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to nonlinear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter.… (More)

This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of the conservation law and a given vector field ensuring that the total variation of the solution along the integral… (More)

A pure-streamfunction formulation is introduced for the numerical simulation of the two-dimensional incompressible Navier–Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacian of the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function… (More)

- Matania Ben-Artzi, Jean-Pierre Croisille, Dalia Fishelov
- SIAM J. Numerical Analysis
- 2006

We study the decay of I (t) = R u(:; t) where u is a nonnegative solution to u t ? u + jruj q = 0 in R n R + with n 1. If 1 q n+2 n+1 then I (t) ! 0 if I (0) < 1 and inf t>0 I (t) > 0 if q > n+2 n+1 .

- Matania Ben-Artzi, Jean-Pierre Croisille, Dalia Fishelov
- SIAM J. Scientific Computing
- 2008

We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [34] as well as in the case of a new fourth order scheme, which is discussed in this paper. It is based on the capacitance matrix method ([10], [8]). The discrete biharmonic operator is… (More)

Two-dimensional deep water waves and some problems in nonlinear optics can be described by various third order dispersive equations, modifying and generalizing the KdV as well as nonlinear Schrödinger equations. We classify all third order polynomials up to certain transformations and study the pointwise decay for the fundamental solutions, Z

- Matania Ben-Artzi, Jiequan Li, Gerald Warnecke
- J. Comput. Physics
- 2006

A direct Eulerian generalized Riemann problem (GRP) scheme is derived for compressible fluid flows. Riemann invariants are introduced as the main ingredient to resolve the generalized Riemann problem (GRP) directly for the Eulerian formulation. The crucial auxiliary Lagrangian scheme in the original GRP scheme is not necessary in the present framework. The… (More)

Abstract. We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in… (More)

- Matania Ben-Artzi, Jiequan Li
- Numerische Mathematik
- 2007

Abstract. The Generalized Riemann Problem (GRP) for nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP… (More)