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Convergence of approximate solutions of conservation laws
Sebastian Noelle, Michael Westdickenberg
TL;DR
The paper addresses how approximate solutions of conservation laws can be shown to converge despite weak-solution nonuniqueness and nonlinear fluxes. It surveys compactness and weak-convergence tools, presents numerical convergence results based on entropy inequalities and related methods, and identifies unresolved rate and mesh-regularity issues.
Problem
For nonlinear conservation laws, weak solutions may be nonunique, while weak convergence alone may not identify nonlinear fluxes because of oscillations.
Method
The paper combines a survey of compactness, compensated compactness, measure-valued, and kinetic methods with convergence analyses for finite volume numerical schemes using maximum principles and discrete entropy inequalities.
Results
Entropy-consistent numerical approximations can be proved convergent, including higher-order finite volume schemes on unstructured grids for merely L∞ initial data, with kinetic methods supplying strong compactness.
Takeaways & Limitations
Entropy inequalities and compactness tools provide the additional control needed to pass to nonlinear limits and establish convergence of approximate solutions.
Takeaways & Limitations
Convergence analyses often assume regular triangulations, while convergence rates remain limited or conjectural in several settings.
Abstract
from arXiv · showhide
In this paper we consider convergence of approximate solutions of conservation laws. We start with an overview over the historical developments since the 1950s, and the analytical tools used in this context. Then we present some of our own results on the convergence of numerical approximations, discuss recent related work and open problems.
1 Exact and approximate solutions of conservation laws
Nonlinear conservation laws can develop shocks, making weak solutions necessary, while nonuniqueness requires entropy conditions. Convergence of approximate solutions hinges on obtaining strong enough compactness to pass nonlinear fluxes to the limit.
- Nonlinear fluxes can produce finite-time shocks even from smooth initial data, so solutions must be interpreted weakly in the distributional sense.
- Weak solutions for general systems may be nonunique, whereas scalar conservation laws have a well-understood Cauchy problem and unique weak entropy solutions.
- A standard existence strategy regularizes the equation, extracts a convergent subsequence of approximate solutions, and identifies its limit as a weak solution.Vanishing viscosity replaces the conservation law with a parabolic problem involving ε∆uε.
- Uniform L∞ bounds yield weak* subsequential convergence, but nonlinear fluxes may not commute with weak limits because oscillations can occur.Strong local L1 convergence would suffice to identify the weak limit of f(uε) with f(u).
- Entropy conditions select physically relevant weak solutions, and entropy inequalities provide crucial estimates for convergence analysis.
- The paper surveys historical developments and analytical tools, then presents results on numerical convergence, related work, and open problems.
2 Historical remarks on compactness arguments
The paper surveys compactness and convergence tools for approximate conservation-law solutions, from BV and L1 methods to weak convergence, measure-valued, and kinetic approaches. It also connects approximation theory with numerical convergence and highlights the kinetic formulation's role in multidimensional finite-volume schemes.
- Regularity estimates and compactness: Classical compactness proves strong L1_loc convergence from positive regularity, especially BV bounds via Helly’s theorem.One-sided Lipschitz estimates for scalar equations can yield BV bounds, while Glimm’s framework extends BV-based analysis to one-dimensional systems.
- Regularity estimates and compactness: Kruzkov’s L1-contraction establishes uniqueness for scalar weak entropy solutions and yields spatial BV bounds through translation estimates.The contraction applies to entropy solutions with different initial data and, by translation invariance, to shifted solutions.
- Approximation theory: Approximation theory uses stability estimates to derive error bounds and convergence rates for vanishing-viscosity and numerical approximations.Kuznetsov’s framework was later extended to more sophisticated schemes, while Lip′-theory measures distance in a weaker dual-Lipschitz topology.
- Weak convergence methods: For unstructured-grid schemes, weak convergence methods replace regularity-based compactness, using Young measures and compensated compactness to control nonlinear limits.Compensated compactness gives weak continuity of products under div-curl assumptions without strong L2_loc compactness; entropy consistency can force Young measures to become Dirac masses.
- Weak convergence methods: Entropy consistency prevents later oscillations beyond those transported from initial data, so strong initial convergence yields strong convergence at later times.The Young-measure result identifies the limit with the unique entropy solution when the stated integrability, solution, entropy, and initial-data conditions hold.
- Kinetic formulation and velocity averaging: The kinetic formulation enables velocity averaging to obtain regularity and compactness, supporting convergence proofs for multidimensional finite-volume schemes.The paper’s authors used this combination to prove convergence for a class of finite-volume schemes for scalar conservation laws in several space dimensions.
3 Some convergence results for finite volume schemes
The paper establishes convergence of higher-order finite volume schemes by combining discrete maximum principles, entropy inequalities, and compactness arguments, including on irregular unstructured grids.
- 3.1 Discrete entropy inequalities: Finite volume schemes use piecewise polynomial approximations on polygonal grids, with conservative numerical fluxes consistent with the physical flux.The fluxes may satisfy upwinding properties and achieve higher-order accuracy in space and time.
- 3.1 Discrete entropy inequalities: A CFL condition enables a discrete maximum principle, but that principle alone does not ensure that the weak-* limit solves the conservation law.The missing ingredient is a discrete entropy inequality.
- 3.1 Discrete entropy inequalities: Higher-order schemes based on Lax-Friedrichs, Engquist-Osher, and generalized Godunov fluxes satisfy discrete entropy inequalities.The analysis rewrites updates as convex combinations and adapts weights to local wave speeds, allowing larger time steps in some settings.
- 3.2 Convergence and error estimates via L1-contraction: Discrete maximum principles and entropy inequalities yield convergence and error estimates for higher-order schemes on unstructured grids, including with merely L∞ initial data.The convergence proof uses DiPerna’s measure-valued solution theory, while related work treats BV data through approximation theory.
- 3.2 Convergence and error estimates via L1-contraction: Convergence analysis can handle grids whose cells become flat as h → 0, and the influence of grid irregularity on convergence rates can be quantified.The rate at which grid degeneration is permitted is shown to be optimal in an example.
- 3.3 Convergence via kinetic formulation: The kinetic compactness theorem converts bounds on kinetic densities, measures, and residuals into compactness of the transformed sequence in L1.Its regularizing effect requires sufficiently nonlinear fluxes and fails for linearly degenerate problems such as advection equations.
- 3.3 Convergence via kinetic formulation: Applying kinetic compactness to numerical approximations provides the strong L1 compactness needed to pass to limits in nonlinear quantities.The proof decomposes the sequence into a small L1 component and a positively regular component.
4 Related work and open problems
Related work develops error estimates and adaptive finite-volume methods, while newer analyses study relaxation and diffusion-dispersion approximations. Open problems include sharper convergence rates, higher-order behavior near discontinuities, and errors in systems.
- Related work: Kuznetsov’s approximation theory underpins a posteriori error estimates and a fully adaptive implicit finite-volume scheme for scalar convection-reaction-diffusion equations.
- Related work: Kinetic formulation combined with velocity averaging analyzes convergence for scalar approximations generated by relaxation or diffusion-dispersion methods.
- Open problems: The h1/2 rate for Lax-Friedrichs is optimal only for first-order approximations of discontinuous linear advection solutions, while unstructured-grid results reach h1/4.
- Open problems: For strictly convex scalar laws, h log h is expected but supported only by numerical experiments, whereas higher rates are established away from discontinuities for higher-order schemes.
- Open problems: In systems, characteristics crossing numerical shock layers from another family can transmit first-order errors into smooth postshock regions, reducing convergence rates even for higher-order schemes.