Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws (original) (raw)

Vanishing viscosity approximation to hyperbolic conservation laws

Journal of Differential Equations, 2008

We study high order convergence of vanishing viscosity approximation to scalar hyperbolic conservation laws in one space dimension. We prove that, under suitable assumptions, in the region where the solution is smooth, the viscous solution admits an expansion in powers of the viscosity parameter ε. This allows an extrapolation procedure that yields high order approximation to the non-viscous limit as ε → 0. Furthermore, an integral across a shock also admits a power expansion of ε, which allows us to construct high order approximation to the location of the shock. Numerical experiments are presented to justify our theoretical findings.

On the Convergence of a Finite Element Method for a Nonlinear Hyperbolic Conservation Law

Mathematics of Computation, 1987

We consider a space-time finite element discretization of a time-dependent nonlinear hyperbolic conservation law in one space dimension (Burgers' equation). The finite element method is higher-order accurate and is a Petrov-Galerkin method based on the so-called streamline diffusion modification of the test functions giving added stability. We first prove that if a sequence of finite element solutions converges boundedly almost everywhere (as the mesh size tends to zero) to a function u, then u is an entropy solution of the conservation law. This result may be extended to systems of conservation laws with convex entropy in several dimensions. We then prove, using a compensated compactness result of Murat-Tartar, that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of Burgers' equation. We also consider a further modification of the test functions giving a method with improved shock capturing. Finally, we present the results of some numerical experiments.

A Numerical Method for First Order Nonlinear Scalar Hyperbolic Conservation Laws in One Dimension

1986

A numerical method for first order nonlinear scalar hyperbolic conservation laws in one dimension is presented, using an idea by Oafermos. In this paper it is proved that it may be used as a numerical method for a general flux function and a general initial value. It is possible to give explicit error estimates for the numerical method. The error in the method is far smaller than in any other method. The numerical method is illustrated in an example.

A note on the effect of artificial viscosity on solutions of conservation laws

Applied Numerical Mathematics, 1996

We consider central difference schemes with artificial viscosity terms for nonlinear hyperbolic systems. We analyze the influence of artificial viscosity on solutions with shocks of nonlinear hyperbolic problems in one dimension. Both stationary and moving shocks are considered. The analysis shows that for the Euler equations one can obtain well behaved (sharp) shock layers when a scalar viscosity coefficient is used. This is not true for a general system. Numerical computations of Burgers' equation and the Euler equations are presented. They support the results from the linear theory.

New adaptive artificial viscosity method for hyperbolic systems of conservation laws

Journal of Computational Physics, 2012

We propose a new finite volume method for solving general multidimensional hyperbolic systems of conservation laws. Our method is based on an appropriate numerical flux and a high-order piecewise polynomial reconstruction. The latter is utilized without any computationally expensive nonlinear limiters, which are typically needed to guarantee nonlinear stability of the scheme. Instead, we enforce stability of the proposed method by adding a new adaptive artificial viscosity, whose coefficients are proportional to the size of the weak local residual, which is sufficiently large ($D, where D is a discrete small scale) at the shock regions, much smaller ($D a , where a is close to 2) near the contact waves, and very small ($D 4 ) in the smooth parts of the computed solution.

Instability of Finite Difference Schemes for Hyperbolic Conservation Laws

Annals of Mathematics Studies

For strictly hyperbolic systems of conservation laws in one space dimension, the Cauchy problem is well posed, within a class of functions having small total variation. However, when solutions with shocks are computed by means of a finite difference scheme, the total variation can become arbitrarily large. As a consequence, convergence of numerical schemes cannot be proved by establishing a priori BV bounds or uniform L 1 stability estimates. In this paper we discuss this instability, due to possible resonances between the speed of the shock and the mesh of the grid.

Large time behaviour of solutions of scalar viscous and nonviscous conservation laws

Applied Mathematics Letters, 1999

We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first-order conservation law.

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

Communications in Mathematical Physics, 1991

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the selfsimilar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves is 0(t~3 /4 ' +ίf2p+σ )