Neumann boundary condition for a non-autonomous Hamilton-Jacobi equation in a quarter plane

We consider Hamilton-Jacobi equation ut+H(u, ux ) = 0 in the quarter plane and study initial boundary value problems with Neumann boundary condition on the line x = 0. We assume that p → H(u, p) is convex, positively homogeneous of degree one. In general, this problem need not admit a continuous viscosity solution when s → H(s, p) is non increasing. In this paper, explicit formula for a viscosity solution of the initial boundary value problem is given for the cases s → H(s, p) is non decreasing as well as s → H(s, p) is non increasing

Conservation law with discontinuous flux

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Monotonization of flux, entropy and numerical schemes for conservation laws

Using the concept of monotonization, families of two step and k-step finite volume schemes for scalar hyperbolic conservation laws are constructed and analyzed. These families contain the FORCE scheme and give an alternative to the MUSTA scheme. These schemes can be extended to systems of conservation law

Godunov-type methods for conservation laws with a flux function discontinuous in space

Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is presented for two-phase flow in porous media. A consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration.Furthermore, for regular solutions, uniqueness has been shown

Solution of a system of nonstrictly hyperbolic conservation laws

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL 8 functions always but may contain d-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution

Solution of convex conservation laws in a strip

In this paper we consider scalar convex conservation laws in one space variable in a strip D =(x, t): 0 ≤x ≤1,t > 0 and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip

Formula for a solution of u<SUB>t</SUB> + H(u,Du) = g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equation ut + H(u,Du) = g in Rn &#215; R+ with u(x, 0) = u0(x). The Hamiltonian H(s,p) is assumed to be convex and positively homogeneous of degree one in p for each s in R. If H is non increasing in s, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions