Construction of solutions and \(L^{1}\)-error estimates of viscous methods for scalar conservation laws with boundary
type: Item
instance of: scholarly article
title: Construction of solutions and \(L^{1}\)-error estimates of viscous methods for scalar conservation laws with boundary
zbMATH Open document ID: 1121.35079
author: Hongxia Liu
author: Tao Pan
publication date: 2007-06-07T00:00:00Z
review text: The authors study an initial boundary value problem for a scalar conservation law \(u_t+f(u)_x=0\), \(t,x>0\) with strictly convex flux function \(f(u)\). This problem is assumed to admit piecewise smooth entropy solutions. The authors describe the structure of such solutions, in particular they show that new types of waves appear. Based on the structure of entropy solutions the authors derive new \(L^1\)~error estimates for the vanishing viscosity method. It turns out that the error of the viscosity solution to the inviscid one is bounded by \(O(\varepsilon^{1/2})\). In the case when the inviscid solutions has special structure (it does not contain waves of new types) the error estimate is \(O(\varepsilon| \ln\varepsilon| )\) as for the initial value problem.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 5162317
zbMATH Keywords: scalar conservation laws
zbMATH Keywords: initial boundary value problem
zbMATH Keywords: weak entropy solutions
zbMATH Keywords: vanishing viscosity methods
zbMATH Keywords: error estimates
zbMATH Keywords: new types of waves
reviewed by: Evgeniy Panov
MaRDI profile type: Publication
full work available at URL: s10114-005-0638-x
OpenAlex ID: W2058395256
cites work: Global weak solutions to the problem of continuous sedimentation of an ideal suspension
cites work: Control of continuous sedimentation of ideal suspensions as an initial and boundary value problem
cites work: Entropy Boundary Conditions in the Theory of Sedimentation of Ideal Suspensions
cites work: Q4151274
cites work: First order quasilinear equations with boundary conditions
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: Convergence of approximate solutions to conservation laws
cites work: Compactness framework of \(L^p\) approximate solutions for scalar conservation laws
cites work: Existence of the entropy solution for a viscoelastic model
cites work: Boundary conditions for nonlinear hyperbolic systems of conservation laws
cites work: Explicit formula for the solution of convex conservation laws with boundary condition
cites work: Explicit formula for scalar non‐linear conservation laws with boundary condition
cites work: Explicit Formula for Weighted Scalar Nonlinear Hyperbolic Conservation Laws
cites work: Q4868758
cites work: The partial differential equation ut + uux = μxx
cites work: Hyperbolic systems of conservation laws II
cites work: Discontinuous solutions of non-linear differential equations
cites work: Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation
cites work: The Sharpness of Kuznetsov's O(√Δx)L 1 -Error Estimate for Monotone Difference Schemes
cites work: The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws
cites work: Optimal L1-Rate of Convergence for The Viscosity Method and Monotone Scheme to Piecewise Constant Solutions with Shocks
cites work: Viscosity methods for piecewise smooth solutions to scalar conservation laws
cites work: First-orderL1-convergence for relaxation approximations to conservation laws
cites work: Viscous limits for piecewise smooth solutions to systems of conservation laws
cites work: Nonlinear stability of shock waves for viscous conservation laws
cites work: Nonlinear asymptotic stability of viscous shock profiles for conservation laws
cites work: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas
cites work: Nonlinear stability of viscous shock waves
cites work: Propagation of a stationary shock layer in the presence of a boundary
cites work: Asymptotic behavior for scalar viscous conservation laws with boundary effect
cites work: Behaviors of Solutions for the Burgers Equation with Boundary corresponding to Rarefaction Waves
cites work: Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws
cites work: Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary
cites work: Large-time behaviors of solutions to an inflow problem in the half-space for a one-dimensional system of compressible viscous gas
cites work: Asymptotic Behavior of a One-Dimensional Compressible Viscous Gas with Free Boundary
cites work: Convergence to travelling fronts of solutions of the \(p\)-system with viscosity in the presence of a boundary
cites work: On L1 Convergence Rate of Viscous and Numerical Approximate Solutions of Genuinely Nonlinear Scalar Conservation Laws
cites work: Pointwise Error Estimates for Scalar Conservation Laws with Piecewise Smooth Solutions
cites work: Pointwise error estimates for relaxation approximations to conservation laws
cites work: Fractional Rate of Convergence for Viscous Approximation to Nonconvex Conservation Laws
cites work: 𝐿¹-convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary
cites work: Q5581897
cites work: On the piecewise smoothness of entropy solutions to scalar conservation laws
cites work: Q4157095
cites work: A regularity theorem for conservation laws
DOI: 10.1007/S10114-005-0638-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5162317