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Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws

Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan Tadmor

arXiv:1402.0909v2math.NAcs.NA

TL;DR

Entropy solutions for multidimensional conservation laws may be nonunique, and numerical approximations may fail to converge to them. The paper constructs numerical approximations to entropy measure-valued solutions and gives sufficient conditions for convergence as the mesh is refined.

  • Problem

    Entropy solutions for multidimensional systems may be nonunique, and tested numerical approximations may fail to converge to them.

    Source: §1.3.2 · p. 7

  • Method

    The paper represents Young measures as laws of random fields and approximates the solution fields with finite-difference or finite-volume schemes.

    Source: §1.5 · p. 8

  • Results

    The paper gives sufficient conditions for mesh-refined approximations to converge to an entropy measure-valued solution and provides schemes satisfying them.

    Source: §1.5 · p. 8

  • Takeaways & Limitations

    The proposed framework provides a viable numerical approach for multidimensional systems within the entropy measure-valued solution paradigm.

    Source: §1.5 · p. 8

    Rigorous stability results remain sparse, with available results covering measure-valued/strong uniqueness when a classical Lipschitz solution is present.

    Source: §7.2 · p. 47

Abstract

from arXiv · show

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.

1 Introduction

The introduction questions whether entropy solutions and their numerical approximations provide a suitable framework for multidimensional conservation laws, then motivates entropy measure-valued solutions and a numerical convergence framework.

1.1 Mathematical framework

  • Recent counterexamples show that entropy solutions for multidimensional systems need not be unique, raising questions about their suitability as a standard framework [18].Weak solutions are generally nonunique after discontinuities form, motivating additional entropy admissibility criteria.Source: §1.1 · pp. 3–4

1.2 Numerical schemes

  • For systems, rigorous convergence proofs are absent even for finite-volume, finite-difference, and finite-element methods, and entropy stability may not ensure convergence.Convergence and stability results for scalar conservation laws are more developed.Source: §1.2 · p. 4

1.3 Two numerical experiments

  • The Sod shock tube is presented as a benchmark commonly used to demonstrate empirical convergence where rigorous convergence and stability results are lacking.The experiment tests whether that apparent convergence generalizes to other flow configurations.Source: §1.3 · p. 5, §1.3.1 · p. 6

1.3.1 Sod shock tube

  • Sod shock-tube approximations form an L1 Cauchy sequence under refinement, while perturbations converge to the unperturbed solution as ε → 0.Similar refinement convergence was obtained with TeCNO2 and the FISH HLLC scheme.Source: Figure 1.1 · p. 5, §1.3.1 · p. 6, Figure 1.2 · p. 6

1.3.2 Kelvin-Helmholtz problem

  • For the Kelvin–Helmholtz problem, refinement produces ever-smaller structures without an L1 Cauchy sequence, and solutions do not converge to the steady state as ε → 0.Similar non-convergence was reported with TeCNO, FISH, and ALSVID schemes, and in weaker W^-1,p norms.Source: §1.3.2 · p. 7, Figure 1.3 · p. 7, Figure 1.4 · p. 8

1.4 A different notion of solutions

  • The authors argue that entropy solutions may not adequately characterize stable limits of numerical approximations, and focus on entropy measure-valued solutions, or Young measures, to represent limits of oscillatory-function sequences.Source: §1.4 · p. 8

1.5 Aims and scope of the current paper

  • The paper studies EMV-solution existence and stability, proposes a random-field-based approximation procedure with sufficient convergence conditions, and presents examples and numerical experiments.The experiments also investigate stability and other properties of computed EMV solutions.Source: §1.5 · pp. 8–9

2 Young measures and entropy measure valued solutions

The paper represents solutions as Young measures—pointwise probability distributions over the phase space—and defines entropy measure-valued solutions by adding an entropy condition to the measure-valued Cauchy problem. The initial-data conditions constrain measure means, so the paper focuses on atomic initial data while noting the formulation also represents uncertainty in non-atomic data.

2 Young measures and entropy measure valued solutions

  • A Young measure assigns each point in space-time a probability measure over the phase space, and composing it with a continuous function gives that function's expectation.Measurable functions correspond to atomic Young measures concentrated at their values.Source: §2 · p. 9

2.1 The measure valued (MV) Cauchy problem

  • An entropy measure-valued solution satisfies the measure-valued Cauchy problem and, in addition, an entropy inequality.The weak formulation uses initial measure-valued data σ.Source: §2.1 · p. 10
  • The initial conditions constrain the mean of the solution measure, which forces equality of the initial measures only when one is a Dirac mass; this motivates the paper's focus on atomic data.For non-atomic initial data, the paper defers the question of uniqueness and interpretation of the initial condition to forthcoming work.Source: §2.1 · p. 10
  • The formulation can represent uncertainty in initial data by treating its random-field law as a non-atomic initial Young measure.The paper connects this representation to uncertainty quantification for conservation laws.Source: §2.1 · p. 11

3 Well-posedness of EMV solutions

The section establishes scalar existence, demonstrates non-uniqueness for non-atomic initial data, and discusses stability for atomic data. For systems, it proves stability relative to classical solutions under stated conditions, while noting that uniqueness may fail after singularities develop.

3.1 Scalar conservation laws

  • For scalar conservation laws, uniformly bounded initial measure-valued data admit an entropy measure-valued solution.The theorem establishes existence without requiring atomic initial data.Source: §3.1 · p. 11
  • Two scalar Burgers random fields with the same initial Young measure can yield distinct entropy measure-valued solutions, despite strong convergence to the initial measure as t → 0 for x ≠ 0.The fields have different two-point correlations, particularly across the origin.Source: §3.1 · p. 12
  • The section identifies stability for scalar problems with atomic initial data, contrasting it with possible non-uniqueness for non-atomic data.The paper proposes a weaker stability notion for systems based on these scalar results.Source: §3.1 · p. 13, §3.2 · p. 14

3.2 Systems of conservation laws

  • For systems, an entropy measure-valued solution supported in a compact set with uniformly convex entropy is controlled relative to a classical solution; atomic initial data then yields that solution uniquely.The result applies while the classical solution exists, establishing consistency with classical solutions.Source: §3.2 · p. 14
  • A classical solution gives a locally unique entropy measure-valued solution, but uniqueness can break down after that solution develops singularities.The section also limits its broader stability analysis to selected systems and numerical experiments.Source: §3.2 · pp. 14, 16

4 Construction of approximate EMV solutions

The section gives a constructive procedure for approximate entropy measure-valued (EMV) solutions, with convergence results under stated conditions. It uses these approximations to compute space-time averages and their statistics.

4 Construction of approximate EMV solutions

  • The procedure constructs approximate EMV solutions and proves their convergence, providing a constructive existence proof for generic systems of conservation laws.The procedure is implemented in the paper’s numerical simulations.Source: §4 · p. 16

4.1.1 Numerical schemes for one- and multi-dimensional conservation laws

  • The numerical schemes use consistent, locally Lipschitz fluxes and exact time integration; the multidimensional construction uses a fixed mesh ratio.The exposition describes one-dimensional schemes and extends the framework to two dimensions.Source: §4.1.1 · pp. 16–17

4.1.2 Weak-∗convergent schemes

  • The algorithm evolves each realization of an initial random field with the numerical scheme and defines the approximate measure-valued solution as the law of the evolved field.These laws serve as approximations to EMV solutions.Source: §4.1.2 · p. 18
  • Under entropy-stability and initial-data consistency conditions, approximate Young measures converge weak* along a subsequence to an EMV solution in one and two space dimensions.The construction also extends to three spatial dimensions.Source: §4.1.2 · pp. 18, 20

4.1.3 Weak-∗convergence with atomic initial data

  • For atomic initial data, the method perturbs the initial state randomly and jointly refines the mesh and perturbation amplitude; a subsequence converges to an EMV solution with the original atomic data.The perturbation is defined as u0(x) + εX(ω, x).Source: §4.1.3 · p. 21

4.2 What are we computing? Weak* convergence of space-time averages

  • Weak* convergence lets the method compute statistics of space-time averages, including the mean and variance, which are observable quantities of interest.The mean uses g(ξ) = ξ and the variance uses g(ξ) = ξ ⊗ ξ.Source: §4.2 · pp. 21–22

4.2.1 Monte Carlo approximation

  • Monte Carlo sampling approximates phase-space integrals, with almost-sure convergence along a subsequence and interchangeable limits in sample size and mesh refinement.The limits are uniform in mesh size.Source: §4.2.1 · pp. 22–23

5 Examples of weak* convergent numerical schemes

The section identifies scheme classes whose approximations satisfy the paper’s convergence criteria for entropy measure-valued (EMV) solutions. Scalar monotone schemes converge strongly under a total-variation condition, while two classes of schemes are presented for systems.

5.1 Scalar conservation laws

  • For scalar laws, monotone schemes satisfy the abstract criteria through uniform boundedness, a discrete entropy inequality, and a weak BV estimate; their approximate Young measures converge strongly when initial-data total variation is uniformly bounded.The total-variation condition can be relaxed to integrability over the random parameter.Source: §5.1 · pp. 23–24

5.2 Systems of conservation laws

  • For systems, the section presents two scheme classes that satisfy the criteria for convergence to EMV solutions.These are TeCNO finite-difference schemes and shock-capturing space-time DG schemes.Source: §5.2 · p. 24, §5.2.1 · p. 25, §5.2.2 · p. 25

5.2.1 TeCNO finite difference schemes

  • TeCNO schemes are formally p-th order accurate, entropy stable, and weakly bounded in variation; if bounded in L∞, their approximate Young measures converge to an EMV solution.The result applies to the multidimensional version as well as the one-dimensional setting described in the section.Source: §5.2.1 · pp. 24–25

5.2.2 Shock capturing space time Discontinuous Galerkin (DG) schemes

  • Shock-capturing streamline-diffusion space-time DG methods satisfy a discrete entropy inequality and a suitable weak BV bound, and were shown to converge to an EMV solution.The section notes that these methods are fully discrete, unlike semi-discrete finite-difference schemes.Source: §5.2.2 · p. 25

6 Numerical Results

The numerical experiments examine mesh refinement and vanishing perturbations for Kelvin–Helmholtz data. Single-sample approximations need not converge, while statistical quantities and approximate Young measures show convergence in the reported tests.

6.1.1 Lack of sample convergence

  • For Kelvin–Helmholtz data, single-sample density approximations do not appear to converge under mesh refinement: finer grids form smaller-scale structures and successive-grid L1 differences do not tend to zero.The computations use the second-order entropy-stable TeCNO2 scheme.Source: §6.1.1 · p. 26

6.1.2 Convergence of the mean and variance

  • With 400 Monte Carlo samples, the Kelvin–Helmholtz density mean forms a Cauchy sequence under mesh refinement, while the variance also appears to converge.Small-scale structures are averaged out in the mean; the variance is higher near the initial mixing layer.Source: §6.1.2 · p. 27

6.1.3 Strong convergence to an EMV solution

  • Successive-mesh Wasserstein differences decrease to zero, indicating strong convergence of the approximate Young measures in space-time and phase space.The pointwise differences are concentrated at interface mixing layers, where the variance is also concentrated.Source: §6.1.3 · p. 31

6.2 Kelvin-Helmholtz: vanishing variance around atomic initial data (ε ↓0)

  • As ε decreases toward zero, single-sample approximations still do not converge, but the sample mean and variance form Cauchy sequences.The limiting mean differs from the steady state, and the limiting variance is non-zero, suggesting a non-atomic EMV solution despite atomic initial data.Source: §6.2 · pp. 32–33
  • At fixed mesh resolution, empirical density PDFs spread the initial point masses over density values between 1 and 2 by t = 2, and appear to converge as the mesh is refined.The authors note that some noise persists on finer meshes, most likely because of the low Monte Carlo sample count.Source: §6.2 · p. 33

6.3 Richtmeyer-Meshkov problem

The Richtmeyer–Meshkov experiments and Kelvin–Helmholtz stability tests show a contrast between nonconvergent single samples and convergent statistics. The computed means and variances are also similar across the tested schemes and perturbations.

6.3.1 Lack of sample convergence

  • For the Richtmeyer–Meshkov problem, single-sample density approximations do not form a Cauchy sequence under mesh refinement.At t = 4, the reentrant shock interacts with the interface and produces small-scale eddy-like structures.Source: §6.3.1 · p. 36

6.3.2 Convergence of the mean and the variance

  • For Richtmeyer–Meshkov data, the mean and variance converge under mesh refinement; the mean retains large-scale shocks and mixing regions while averaging out small-scale features.The variance is concentrated at shocks and especially in the mixing layer around the original interface.Source: §6.3.2 · pp. 36, 38

6.4 Measure valued (MV) stability

  • The stability tests assess the weaker measure-valued notion because EMV solutions need not be unique or stable for general measure-valued initial data.The paper describes MV stability as stability with respect to perturbations of atomic initial data.Source: §6.4 · p. 38

6.4.1 Stability with respect to different numerical schemes.

  • At ε = 0.01, TeCNO2, TeCNO3, and FISH produce similar means and variances, indicating MV stability across the tested numerical discretizations.The authors note uncertainty about whether FISH satisfies the discrete entropy inequality or weak BV bound.Source: §6.4.1 · pp. 38, 41

6.4.2 MV stability with respect to different perturbations

  • For small ε, phase and amplitude perturbations yield very similar means and variances; four tested initial perturbation sets also produce similar computed statistics.The reported comparisons include uniform and standard-normal choices for the perturbation coefficients.Source: §6.4.2 · pp. 41–42

7 Discussion

The discussion argues that mesh refinement can produce nonconvergent single realizations even as ensemble statistics and approximate Young measures converge. It presents EMV solutions as a broader framework, while noting unresolved questions about stability, uniqueness, and computational cost.

7.1 What do the numerical experiments tell us

  • Generic multidimensional systems lack known global existence and uniqueness results for entropy solutions, and examples have infinitely many such solutions [18].The numerical evidence also suggests entropy solutions may inadequately describe persistent fine-scale flow structures.Source: §7 · p. 42, §7.1 · p. 45
  • Mesh refinement produces increasingly fine structures without observed convergence for single realizations in two-dimensional Euler examples.The paper says the approximations do not seem to form a Cauchy sequence in L1.Source: §7.1 · pp. 45–46
  • The proposed EMV procedure evolves random fields with entropy-stable schemes; discrete entropy inequalities and space-time weak BV bounds are among sufficient convergence criteria.The stated criteria also include a uniform L∞ bound, described as a technical assumption.Source: §7.1 · p. 45
  • Ensemble means and variances converge under refinement, and approximate Young measures appear to converge strongly to an EMV solution.The paper describes this strong convergence as stronger than convergence of statistical quantities alone.Source: §7.1 · p. 46
  • Computed solutions from atomic initial data need not remain atomic, and the solution operator can spread the initial measure's support.The Kelvin-Helmholtz case is especially striking because an unstable atomic entropy measure-valued solution exists for its steady-state data.Source: §7.1 · p. 46

Stability

  • Different schemes and perturbations of atomic initial data appear to converge to the same EMV solution, suggesting stability of the approximation procedure.The authors propose measure-valued stability as a possible framework for discussing EMV-solution stability.Source: Stability · pp. 46–47

7.2 Issues for future investigation

  • Uniqueness and stability remain open concerns; the authors suggest that additional admissibility criteria may be needed even for measure-valued stability.They also identify development of other schemes satisfying their criteria as a future direction.Source: §7.2 · p. 47
  • Monte Carlo evaluation of phase-space integrals can be computationally expensive because convergence with sample count is slow.The authors identify multilevel Monte Carlo, stochastic collocation, and stochastic Galerkin methods as possible alternatives to investigate.Source: §7.2 · p. 47

A Young measures

This short introduction points readers to references for the theory of Radon and probability measures and Young measures.

A Young measures

  • The section directs readers to references for Radon and probability measures and for Young-measure theory.It describes itself as a very short introduction.Source: A Young measures · p. 47

A.1 Probability measures

This appendix section introduces finite Radon measures and probability measures, along with weak* convergence and Wasserstein distance. It also outlines a sampling-based way to approximate discrete probability distributions.

A.1 Probability measures

  • Finite Radon measures are identified with the dual of C0(RN), and weak* convergence is defined through convergence of pairings with every function in C0(RN).The section also notes that weak* convergence is called weak or vague convergence.Source: §A.1 · pp. 47–48
  • For probability measures with finite pth moments, the Wasserstein metric is defined using joint probability measures with the given marginals and metrizes weak convergence.The passage specifies this property for measures in Pp(RN).Source: §A.1 · p. 48
  • Discrete distributions formed from independent samples converge weak* almost surely to their source distributions, motivating a sorted-sample algorithm for computing their Wasserstein distance.The passage describes the algorithm as easy to implement.Source: §A.1 · p. 48

A.2 Young measures

The appendix defines Young measures as measurable maps into probability measures and distinguishes weak* from strong convergence. It also states a compactness result for sequences satisfying a no-mass-leakage condition.

A.2 Young measures

  • A Young measure maps each point of its domain to a probability measure, with measurability defined through pairings against functions in C0(RN).The notation νz denotes the probability measure assigned at z.Source: §A.2 · p. 49
  • A Young measure is uniformly bounded when all its measures are supported in one compact set; atomic Young measures correspond to measurable functions.For an atomic measure ν = δu, uniform boundedness is equivalent to u being L∞ bounded.Source: §A.2 · p. 49
  • The section distinguishes weak* and strong convergence for Young measures and states that strong convergence of atomic measures corresponds to convergence of the underlying functions in Lp.The supplied passages define the two topologies but do not give their full displayed criteria.Source: §A.2 · p. 49
  • A sequence of Young measures that does not leak mass at infinity has a weak* convergent subsequence, with support control under the stated compact-support condition.The theorem's subsequential limit is a nonnegative measure-valued function.Source: §A.2 · pp. 49–51

A.3 Random fields and Young measures

This appendix connects jointly measurable random fields with Young measures and shows that every Young measure can be realized as the law of a random field.

A.3 Random fields and Young measures

  • A jointly measurable random field defines a Young measure by taking the probability that its value at each spatial point lies in a Borel set.The law is written as ν_z(F) = P({ω : u(ω, z) ∈ F}).Source: §A.3 · p. 51
  • For each continuous function vanishing at infinity, the corresponding expectation is measurable in the spatial variable.This measurability is used in defining the associated Young measure.Source: §A.3 · pp. 51–52
  • The law definition is independent of the representative chosen for a random field when representatives agree almost everywhere on the product space.The resulting laws agree for almost every spatial point.Source: §A.3 · pp. 51–52
  • Every Young measure on D to R^N is the law of a Borel measurable random field on a probability space.The probability space can be chosen as [0, 1) with Lebesgue measure.Source: §A.3 · p. 52
  • The construction approximates a Young measure with measurable finite-valued fields whose pointwise limit is measurable and has the prescribed law almost everywhere.The approximants converge pointwise, and the resulting field has law ν_z for almost every z.Source: §A.3 · pp. 52–53

B Proof of Theorem 4.9

The proof uses independent, identically distributed samples to establish convergence of the sample mean to the ensemble average, with almost-sure convergence along a subsequence.

B Proof of Theorem 4.9

  • The sample mean converges to the corresponding ensemble average, and a subsequence converges P-almost surely.The argument uses that the sample-derived variables are independent and identically distributed.Source: B Proof of Theorem 4.9 · pp. 53–54

C Time continuity of approximations

The time integration procedure makes approximate measure-valued solutions time continuous, so they attain the initial data in a certain sense and can be evaluated at specific times. The stated convergence result gives ΨΔx(t) → Ψ(t) as Δx → 0 for almost every t.

C Time continuity of approximations

  • The approximate measure-valued solutions are time continuous, attain initial data in a certain sense, and can be evaluated at a specific time.This follows from the time integration procedure.Source: C Time continuity of approximations · p. 54
  • The stated convergence result gives ΨΔx(t) → Ψ(t) as Δx → 0 for almost every t ∈ [0, T].The functions in the result are generated by Algorithm 4.1.Source: C Time continuity of approximations · p. 54
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