A moving mesh numerical method for hyperbolic conservation laws
TLDR
It is shown that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics.Abstract:
We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approxima- tions, is accurate to O(N-1). These numerical methods for conservation laws are the first to have proven convergence rates of greater than O(fN-1/2). 1. Introduction. It is well-known that the solution of the hyperbolic conservation law,read more
Citations
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Journal ArticleDOI
The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case
TL;DR: The two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods are studied, which can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate.
Book
Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves
Philippe G. LeFloch,J Novotny +1 more
TL;DR: In this article, the Riemann problem is formulated as a class of linear hyperbolic equations, and the entropy dissipation function is defined as a function of the total variation functional.
Journal ArticleDOI
Fully adaptive multiresolution finite volume schemes for conservation laws
TL;DR: The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid.
MonographDOI
Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs
Journal ArticleDOI
On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients
TL;DR: In this article, the authors study nonlinear degenerate parabolic equations where the flux function $f(x,t,u) does not depend Lipschitz continuously on the spatial location of the initial value.
References
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High resolution schemes for hyperbolic conservation laws
TL;DR: In this article, a class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented, which are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function.
Book
Adaptive mesh refinement for hyperbolic partial differential equations
Marsha Berger,Joseph Oliger +1 more
TL;DR: This work presents an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques based upon Richardson-type estimates of the truncation error, which is a mesh refinement algorithm in time and space.
Book
Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves
TL;DR: Quasi-linear Hyperbolic Equations Conservation Laws Single Conservation Laws The Decay of Solutions as t Tends to infinity Hypothesis of conservation laws Pairs of Conservation Laws as mentioned in this paper.
Journal ArticleDOI
First order quasilinear equations in several independent variables
TL;DR: In this paper, a theory of generalized solutions in the large Cauchy's problem for the equations in the class of bounded measurable functions is constructed, and the existence, uniqueness and stability theorems for this solution are proved.