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Why many theories of shock waves are necessary. Convergence error in formally path-consistent schemes
Manuel J. Castro, Philippe G. LeFloch, María Luz Muñoz-Ruiz, Carlos Parés
TL;DR
The paper asks whether finite difference schemes for nonconservative hyperbolic systems converge to correct shock-containing weak solutions. It analyzes formally path-consistent schemes and shows that convergence errors can arise, while vanishing for systems with linearly degenerate nonconservative fields.
Problem
Finite difference schemes for nonconservative hyperbolic systems must be assessed for convergence toward correct weak solutions containing shock waves.
Method
The paper combines analysis of convergence errors and path-dependent equivalent equations with numerical studies of fluid-dynamics models.
Results
For linearly degenerate nonconservative fields, the convergence error measure vanishes and formally path-consistent schemes converge to exact solutions.
Takeaways & Limitations
Formally using the same family of paths does not guarantee correct shock solutions for general systems, although the error can be negligible except on fine meshes, large shocks, or long simulations.
Takeaways & Limitations
Computing the appropriate path family may require regularized shock profiles, while rigorously convergent Glimm and front-tracking methods require explicit Riemann solvers.
Abstract
from arXiv · showhide
We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat's theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. We generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, a "convergence error" source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. We discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several models arising in fluid dynamics. For systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For some other models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration.
1. Introduction
The paper examines whether finite difference schemes for nonconservative hyperbolic fluid models converge to correct shock-containing weak solutions. It motivates path-based weak-solution theory and direct numerical analysis because solutions depend on regularization and different schemes may select different limits.
- 1. Introduction: Nonconservative hyperbolic models arise as simplified descriptions of two-phase and two-layer flows, motivating convergence analysis for shock-containing solutions.
- 1. Introduction: Because these systems are non-divergence form, discontinuous solutions require weak-solution concepts beyond distributional solutions.
- 1. Introduction: Different regularization mechanisms or approximation schemes may converge to different solutions, so viscosity, capillarity, and relaxation effects matter for well-posedness.
- 1. Introduction: Shock-wave Rankine-Hugoniot relations are determined by traveling-wave solutions of the chosen regularized system.
- 1. Introduction: Simplified models can be stable under regularization, whereas general systems such as full two-phase-flow models require the general DLM theory.
- 1. Introduction: The paper therefore uses direct discretization of nonconservative models and examines numerical strategies for their approximation.
2. The convergence error measure
Formally path-consistent finite difference schemes can converge to a limiting nonconservative system with a measure-valued source term, called the convergence error. Under stronger graph-convergence conditions, this error vanishes and the limit is a weak solution.
- DLM families of paths and nonconservative products: The DLM framework defines nonconservative products for BV functions only after prescribing a family of Lipschitz continuous paths in phase space.The resulting product is a bounded measure and agrees with the distributional derivative in the conservative case.
- A class of finite difference schemes: The paper studies a general class of finite difference schemes, including Godunov, Roe, and Lax-Friedrichs methods, that are formally consistent with a fixed path family.Formal path consistency generalizes conservative-scheme consistency but requires careful handling for nonconservative systems.
- Convergence to a nonconservative system with measure-source term: For uniformly bounded BV approximations, every almost-everywhere convergent subsequence has a bounded-measure convergence error in the limiting hyperbolic system.The scheme is decomposed into a consistent hyperbolic part and an error term, with DLM stability results controlling the limit.
- Convergence to a nonconservative system with measure-source term: If the Φ-completions of the approximate graphs converge uniformly to the Φ-completion of the limit, the convergence error vanishes identically.This graph-convergence condition is strong, holds for Glimm and front-tracking schemes, and usually fails for finite difference schemes.
- Convergence to a nonconservative system with measure-source term: The convergence error can be evaluated through the measure μv or through the Rankine-Hugoniot shock curves associated with each numerical scheme.The paper emphasizes numerical shock-curve computation as a way to assess a scheme’s range of validity.
3. Equivalent equations for nonconservative systems
Equivalent equations expose how numerical schemes regularize nonconservative systems and why formal path consistency does not guarantee convergence to the intended weak solutions. For systems, the equivalent equation depends explicitly on the prescribed path family.
- Derivation of the equivalent equation: The equivalent equation for the generalized Lax-Friedrichs scheme is derived by formal Taylor expansion through second and third order.The derivation rewrites time derivatives using spatial derivatives to obtain modified equations.
- Derivation of the equivalent equation: The path family enters the modified equation only through the term I2(v), distinguishing nonconservative systems from conservative equations.The displayed higher-order terms include derivatives of the matrix A and path derivatives with respect to the left and right states.
- Derivation of the equivalent equation: The equivalent-equation calculation is highly complex for nonconservative schemes.This complexity is identified as a central difficulty of the equivalent-equation approach in the nonconservative setting.
- Relation with vanishing-viscosity solutions: When weak solutions are selected by parabolic regularization, the numerical viscous terms can correspond to a different regularization than the prescribed one.Unlike the conservative case, vanishing-viscosity limits depend on the regularization used.
4. Examples of nonconservative hyperbolic systems
The paper illustrates several nonconservative fluid-dynamics systems, defining their characteristic structures, path-dependent jump conditions, and numerical approximation requirements. The examples include balance laws, shallow-water flow, and two-layer systems with explicit hyperbolicity constraints.
- A class of systems of balance laws: A representative nonconservative system uses prescribed paths formed from characteristic integral curves and line segments to define shock jump conditions.For a sample Riemann problem, the path combines a 1-rarefaction arc with segments associated with a 2-shock.
- A class of systems of balance laws: For systems with a linearly degenerate field, families satisfying the stated path restrictions yield the same weak-solution notion.These solutions include shocks across which the auxiliary variable is continuous and stationary contact discontinuities along the linearly degenerate field.
- A class of systems of balance laws: Shock waves where the auxiliary variable is continuous are captured independently of the path choice, whereas stationary contacts require schemes that preserve the relevant equilibrium structure.The required numerical condition is also identified as necessary for well-balanced schemes.
- A class of systems of balance laws: The balance-law examples include shallow water flow in a straight channel, with variables representing layer thickness, mass flow, velocity, gravity, and topography.The model has characteristic speeds u − c, u + c, and 0, with c = √gh.
- Two-layer shallow water system: The two-layer shallow-water model has four characteristic fields, but its internal eigenvalues can become complex, causing loss of hyperbolicity and limiting the model’s validity.The paper restricts numerical experiments to states for which the system matrix has real eigenvalues and is strictly hyperbolic.
- Two-layer shallow water system: For the Roe scheme, numerical Hugoniot curves converge toward a limit that differs from the exact Hugoniot curve.The exact curve is shown continuously and the numerical curve with dots in Figure 2.
5.1. A simplified system
The simplified system shows that formally path-consistent Roe and Godunov schemes can converge to limits that differ from the exact weak solution. Numerical tests also compare shock resolution and conservation behavior for Godunov and Glimm methods.
- Hugoniot curves: Roe and Godunov are compared against the exact Hugoniot curve using distinct line and marker styles.The exact curve is shown in red, Roe in blue with squares, and Godunov in green with circles.
- Hugoniot curves: The numerical Hugoniot curves for Roe and Godunov converge, but their limit is not the exact Hugoniot curve.The schemes are formally consistent with prescribed path families, yet the limiting shock relation differs from the exact one.
- Riemann problems: The exact Riemann solution is a 1-shock, but Godunov introduces a 2-rarefaction in the computed solution.The comparison uses Godunov and Glimm at t = 0.5 with ∆x = 0.001 and CFL=0.5.
- Conservation: Godunov satisfies the conservation property for the integral of h exactly, whereas the figure compares its evolution with Glimm and the exact value.The conservation property follows from the first equation being a conservation law.
5.2. Shallow water system
For the shallow water system, shock waves over continuous bottoms are captured correctly, while stationary contacts at bottom discontinuities require paths satisfying (R1). Well-balanced path choices exactly capture the stationary contact.
- 5.2.2. Stationary contact discontinuities: Stationary contact discontinuities at jumps of the bottom are correctly captured only when the path family satisfies (R1).For continuous bottom functions, the shock waves are correctly captured even with a simple path family.
- 5.2.1. Dam-break problem: Roe and modified Lax-Friedrichs schemes converge to the same dam-break solution for continuous bottom topography.The computation uses meshes with 800, 1600, and 3200 cells at final time t = 0.6.
- 5.2.2. Stationary contact discontinuities: With the family of segments, both schemes converge to the same discontinuous function rather than the exact stationary-contact solution.The tested states belong to the same integral curve, with hr approximately 0.7892441190408083.
- 5.2.2. Stationary contact discontinuities: Using paths satisfying (R1) for every linearly degenerate integral curve, the stationary contact discontinuity is exactly captured.The path consists of an integral-curve arc followed by a segment on a plane H = const.
5.3. Two-layer shallow water system
In the homogeneous two-layer shallow water system, numerical approximations can converge without yielding weak solutions for the prescribed paths. External shocks are captured more closely than internal shocks in the reported comparisons.
- 5.3. Two-layer shallow water system: Roe and Lax-Friedrichs approximations converge but do not converge to weak solutions, even when the scheme and jump conditions use the same path family.This behavior is established for the homogeneous system with H = cst.
- 5.3.1. Approximation of internal shocks: For internal shocks, numerical Hugoniot curves converge to a limit that is not a weak solution for the chosen paths.The exact and numerical curves are very close when the states are sufficiently close or the shock speed is near zero.
- 5.3.2. Approximation of external shocks: Numerical viscosity is stronger for internal shocks, so external shocks are expected to be better captured by Lax-Friedrichs or Roe schemes.The CFL condition adjusts the numerical velocity to the external eigenvalues.
- 5.3.2. Approximation of external shocks: For the external-shock test, the exact, Lax-Friedrichs, and Roe Hugoniot curves are much closer than in the internal-shock test.The comparison uses a mesh with 2000 cells.
5.4. Influence of the family of paths
Changing the family of paths affects the weak and numerical solutions differently across schemes. Lax-Friedrichs Hugoniot curves remain unchanged under the tested path perturbations, whereas Roe curves depend on the perturbation.
- 5.4. Influence of the family of paths: The study varies the path family Φϵ using ϵ ∈ {0.0, 0.01, 0.02, 0.03, 0.04, 0.05}.These path choices are used to examine their influence on weak and numerical solutions.
- Lax-Friedrichs scheme: For Lax-Friedrichs, all numerical Hugoniot curves coincide across the tested ϵ values.The curves are reparameterizations of the same curve, consistent with an ϵ-independent second-order modified equation.
- Roe scheme: For Roe, the numerical Hugoniot curves depend on ϵ and do not converge to the weak solutions associated with the chosen paths.As before, exact and numerical curves become close when the states are near each other or the shock speed is near zero.
6. Conclusions
The paper recommends formally path-consistent finite-difference schemes while identifying their limitations and convergence errors. It finds vanishing errors for linearly degenerate fields and proposes shock curves for assessing scheme validity in more general models.
- A finite-difference scheme should follow a physics-consistent regularization, its DLM family of paths, and a scheme converging to weak solutions associated with those paths.Computing the path family may require regularized shock profiles, making the strategy difficult in practice.
- Directly discretizing a simplified nonconservative model can be preferable when solving its more complex conservative counterpart would be costlier.The two-layer shallow water system is given as an example.
- Formally path-consistent schemes preserve the DLM nonconservative-product definition and are conservative on any conservative subsystem.Their captured shocks also correspond to a regularization with higher-order terms vanishing as ∆x tends to 0, although that regularization depends on the paths and scheme.
- The convergence error is noticeable only for very fine meshes, large-amplitude discontinuities, or long-time simulations, and shock curves provide a way to assess scheme validity.For more general models, the measure can be evaluated accurately by plotting each scheme’s shock curves; two-layer shallow-water shocks have also agreed well with experimental measurements.
- For linearly degenerate nonconservative fields, the convergence error vanishes and formally path-consistent schemes correctly approximate all discontinuities.In this special case, the schemes converge to exact solutions.