One-dimensional Scalar Conservation Laws Research Articles

A MUSCL method satisfying all the numerical entropy inequalities

We consider here second-order finite volume methods for one-dimensional scalar conservation laws. We give a method to determine a slope reconstruction satisfying all the exact numerical entropy inequalities. It avoids inhomogeneous slope limitations and, at least, gives a convergence rate of Δx l/2 . It is obtained by a theory of second-order entropic projections involving values at the nodes of the grid and a variant of error estimates, which also gives new results for the first-order Engquist-Osher scheme.

Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity

It is shown that as time approaches infinity, the solution of the initial value problem for a regularized one-dimensional scalar conservation law converges along rays to the solution of a certain Riemann problem for the hyperbolic conservation law, even when this conservation law is not genuinely nonlinear.