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Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen

arXiv:0901.0816v1math.APmath.NA

TL;DR

The paper addresses degenerate doubly nonlinear hyperbolic-parabolic equations whose nonlinearities may be non-Lipschitz, making standard numerical treatment and compactness arguments difficult. It constructs DDFV schemes in two and three dimensions, establishes their discrete properties and solvability, and proves strong convergence of their approximations to the unique entropy solution. The analysis also identifies scope constraints imposed by the conformal mesh conditions.

  • Problem

    The target equations combine nonlinear convection, doubly nonlinear diffusion, strong degeneracy, shocks, and non-Lipschitz fluxes, for which convergent numerical schemes were not known in the stated generality.

  • Method

    The paper develops two- and three-dimensional DDFV schemes with discrete duality and entropy dissipation properties, using compactness and monotonicity arguments to analyze their limits.

  • Results

    The discrete problems admit solutions, and the associated approximations converge strongly to the unique entropy solution as the mesh size tends to zero.

  • Takeaways & Limitations

    The schemes provide a convergent finite volume treatment for the considered two- and three-dimensional degenerate hyperbolic-parabolic problems, including non-Lipschitz nonlinearities.

  • Takeaways & Limitations

    The studied conformal DDFV schemes require orthogonality, Delaunay, and additional mesh conditions, and conformity removes the simple local refinement advantage of 2D schemes.

Abstract

from arXiv · show

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès \cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" $L^\infty$ weak-$\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz nonlinearities.

1. Introduction

The paper studies degenerate hyperbolic-parabolic equations with potentially non-Lipschitz nonlinearities and develops convergent DDFV schemes in two and three dimensions. The setting combines nonlinear convection, doubly nonlinear diffusion, degeneracy, shocks, and homogeneous Dirichlet conditions.

  • Problem setting: The model includes degenerate hyperbolic-parabolic equations with continuous nonlinearities that need not be locally Lipschitz.The flux and diffusion functions are normalized, while the diffusion function is continuous and nondecreasing.
  • Problem setting: These problems encompass conservation laws, heat and porous medium equations, degenerate convection-diffusion equations, p-Laplace equations, and elliptic-parabolic equations.The paper also notes applications of degenerate parabolic equations to porous-media flow and sedimentation-consolidation.
  • Numerical approach: Finite volume methods are proposed because the equations are written in divergence form and finite volume discretization is well suited to such problems.The paper’s chief numerical goal is to construct and analyze a specific finite volume scheme class.
  • Numerical approach: The generality of the target problems combines nonlinear convection, doubly nonlinear diffusion, strong degeneracy, shocks, and discontinuous entropy solutions.Without Lipschitz continuity of the convective flux, the CFL condition does not apply, motivating an implicit treatment of convection.
  • Numerical approach: The paper constructs discrete duality finite volume schemes in two and three spatial dimensions, including new three-dimensional schemes with discrete duality properties.The schemes use established approximations for convection and diffusion, while the three-dimensional construction introduces new DDFV designs.

2. Notions of solution and well-posedness

The paper formulates entropy and double-process solution frameworks to handle shocks, degeneracy, boundary conditions, and weak compactness. It proves existence, uniqueness, and well-posedness, with double-process solutions collapsing to a single entropy solution.

  • Entropy framework: Semi-Kruzhkov entropy pairs encode both interior entropy conditions and the homogeneous Dirichlet boundary condition.The framework is needed because weak solutions may be nonunique without additional entropy inequalities.
  • Entropy framework: Entropy double-process solutions use two process functions to represent the two approximations produced by DDFV schemes.The two functions are a technical device for handling weak compactness of paired discrete approximations.
  • Entropy framework: The double-process formulation is equivalent to the entropy formulation when both process functions are independent of the process parameter and equal the same function.An entropy solution generates such a triplet, and conversely the identified common function is an entropy solution.
  • Existence and identification: An entropy double-process solution exists with identical process functions, and the associated compactness argument yields strong convergence of the transformed diffusion variable.The proof obtains strong L1 convergence of the transformed variable, weak convergence of gradients and fluxes, and then identifies the limiting diffusion structure.
  • Existence and identification: The limiting diffusion flux is identified as a(∇A(u)), while strict monotonicity and equi-integrability improve the gradient convergence to strong Lp convergence.The entropy inequalities then pass to the limit, producing an entropy double-process solution with equal process functions.
  • Well-posedness: The entropy double-process solution is unique and reduces to a unique entropy solution, yielding L1 comparison and order-preservation for initial data and source terms.Equal initial data and source terms imply equality of the corresponding entropy solutions.

3. Discrete duality finite volume (DDFV) schemes

The paper constructs conformal primal–dual finite-volume meshes in two and three dimensions, organizing them into diamonds and subdiamonds for the DDFV discretization. Mesh geometry and condition (19) are essential for the entropy inequalities and convergence analysis.

  • Mesh construction: The meshes are assumed to be triangles or tetrahedra satisfying a Delaunay condition and additional geometric conditions.In three dimensions, each tetrahedral face contains the centre of its circumscribed circle.
  • Geometric condition: Condition (19) is unnecessary to define the scheme but is required for discrete entropy inequalities and convergence.Dropping it can require signed subdiamond volumes; the resulting discrete calculus remains consistent, but entropy dissipation inequalities may fail.
  • Mesh construction: DDFV meshes use a primal partition and a Voronoi dual partition whose centres and vertices are mutually identified.The mesh is built from control volumes, dual control volumes, interfaces, and neighbouring cells.
  • Discrete operators: Diamonds and subdiamonds support the discrete gradient and divergence operators used to discretize the second-order diffusion operator.The construction uses geometric decompositions associated with primal and dual interfaces.
  • Diamonds and subdiamonds: Neighbouring primal and dual cells define diamonds, with each three-dimensional diamond split into three subdiamonds.In two dimensions, every diamond is itself a subdiamond.

3.2. Mesh parameters and regularity of meshes.

The scheme quantifies mesh regularity and reconstructs discrete gradients and divergences on diamonds and subdiamonds. Conformity, bounded regularity, and exactness on affine functions provide the geometric foundation for the DDFV analysis.

  • Mesh regularity: The mesh regularity constant controls the relative dimensions and proportions of neighbouring control volumes, diamonds, and dual control volumes.All convergence results require mesh families with uniformly bounded regularity constants.
  • Discrete operators: Discrete gradients are reconstructed on diamonds, while discrete divergences are assembled on subdiamonds using primal and dual interface geometry.Subdiamond fields inherit the corresponding diamond field, and normal orientations enter the divergence formulas.
  • Conformity: The conforming double-mesh property is especially important in the L1 framework for degenerate diffusion and hyperbolic convection.Without the convective term, non-conformal double meshes can be treated in a variational framework.
  • Discrete gradient: The discrete gradient is exact on affine functions.For an affine function w(x)=w0+r·x, the reconstructed discrete gradient equals the constant vector r.
  • Gradient reconstruction: In two dimensions, gradient reconstruction uses orthogonal primal and dual interface directions; in three dimensions, it combines weighted projections associated with the primal face and dual edges.The three-dimensional formula relies on a reconstruction result in the plane containing the primal interface.
  • Dimensional scope: The direct reconstruction formula does not generally extend to dimensions d≥4 except for specially structured meshes.Uniform simplicial meshes are cited as an example where such higher-dimensional extensions may hold.

3.4. Penalization operator.

The penalization operator couples the two components of the DDFV approximation of A(u), ensuring that both converge to the same limit. Its specific form is not essential for convergence, and alternative choices remain possible.

  • Construction: The operator acts on the difference between the primal and dual discrete values of the approximated quantity.
  • Purpose: The penalization operator is introduced to ensure that the two components of the discrete double function converge to the same limit.This avoids relying on strong convergence of the discrete gradient.
  • Variants: The proposed penalization is only the simplest choice, since power-type replacements and different powers of mesh size would preserve convergence.The optimal choice of penalization operator is left open.

3.5. Discrete convection operator.

The discrete convection operator uses numerical fluxes on each of the two meshes, with consistency, conservativity, monotonicity, and continuity properties adapted to continuous non-Lipschitz fluxes.

  • Flux construction: Discrete convection is defined separately on the primal and dual meshes using numerical flux functions across neighboring interfaces.This follows the finite-volume construction used to approximate f(u)·ν across each interface.
  • Flux properties: The numerical flux is consistent with the physical flux and antisymmetric under exchanging neighboring cells.The construction also requires monotonicity in its arguments.
  • Non-Lipschitz fluxes: The flux functions share the modulus of continuity of the continuous convective flux rather than requiring a common Lipschitz constant.This adapts standard flux assumptions to general continuous f.
  • Examples: Godunov, Lax-Friedrichs, Engquist-Osher, and Rusanov fluxes are cited as examples satisfying the numerical-flux framework.
  • Operator: The discrete convection operator is defined on discrete functions through the separately constructed primal- and dual-mesh fluxes.
  • Projection and test functions: The mesh projection operator maps L1(Ω) functions to discrete functions, while projected gradients and boundary values provide discrete test-function data.For test functions vanishing on the boundary, the associated boundary values are zero; compact support gives the corresponding dual-boundary property when the mesh is sufficiently fine.

3.7. Dependency on t and further notation. •

The scheme is parameterized by the maximum of the mesh size and time step, with discrete functions and fields indexed over time levels and primal-dual mesh entities. The finite-volume problem is posed at each time step with boundary and initial conditions.

  • Discretization parameter: The discretization parameter is h = max{size(T), ∆t}, and convergence is studied as h decreases to zero.
  • Time discretization: The number of time steps is N, the integer part of T/∆t, and the dependence of N, T, and ∆t on h is usually suppressed.
  • Discrete functions: A discrete function on the space-time domain is a sequence of spatial discrete functions indexed by time level.
  • Discrete notation: Discrete fields and functions are defined on both primal and dual mesh entities and can be composed with mappings such as A or ϕ.Nonnegativity is defined entrywise over all primal and dual mesh values.
  • Discrete problem: At each time step, the finite-volume problem seeks a discrete function satisfying the scheme equations together with boundary and initial conditions.The vector used in the scheme is constructed from values of A(u) on the primal and dual meshes.
  • Scheme components: The explicit scheme formulation introduces numerical convection fluxes, diffusion-related quantities, and discrete gradients of A(u).

4. Elements of discrete calculus for DDFV schemes

The DDFV calculus provides discrete summation-by-parts, duality, chain-rule, and entropy-dissipation tools for analyzing the diffusion and convection terms. These identities rely on mesh conformity and stated positivity or boundary assumptions.

  • Discrete identities: The analysis collects summation-by-parts formulas and chain rules needed for the discrete problem.
  • Diffusion duality: DDFV schemes satisfy a discrete duality property between the negative divergence and gradient operators.This is the finite-volume analogue of continuous divergence-gradient duality.
  • Entropy dissipation: The schemes also satisfy entropy dissipation inequalities for nondecreasing functions under the stated positivity and boundary assumptions.The hypotheses include either θ(0)=0 or a sufficiently small mesh with a suitable test function.
  • Assumptions: The discrete identities require conformity of the meshes, the specified form of the coefficient a, and, in three dimensions, condition (19).
  • Boundary terms: Boundary contributions are handled by extending the summation formulas to boundary volumes and using the prescribed boundary values.
  • Chain rules: The chain-rule estimates rely on convexity inequalities for the monotone mapping θ and the nonlinear function A.
  • Convection entropy formula: Convective terms admit a separate entropy-dissipation duality formula on the primal and dual meshes.Its proof uses the numerical-flux assumptions, boundary conditions, and summation by parts.

5. Properties of discrete operators and functional spaces

This section establishes consistency, stability, and compactness properties for discrete functions and operators on double meshes. These results identify common weak limits for the two mesh components and support strong convergence when time-translate estimates are available.

  • Consistency and discrete operators: Discrete projection and gradient operators satisfy consistency properties in Lebesgue and Sobolev spaces, while the penalization operator vanishes asymptotically.The estimates are formulated for mesh size and time step jointly through h=max{size(T),∆t}.
  • Functional inequalities and compactness: The discrete framework provides Poincaré-type inequalities, translation estimates, and asymptotic compactness results for uniformly regular double meshes.These properties apply to discrete functions bounded in the relevant discrete W 1,p spaces.
  • Strong compactness: Uniform time-translate estimates yield strong convergence of the discrete approximations in L^p(Q).The compactness result is obtained along a subsequence of meshes as h→0.
  • Identification of weak limits: Weakly convergent discrete gradients are identified with the gradient of the weak limit through discrete integration-by-parts arguments.The limit satisfies w∈L^p(0,T;W_0^{1,p}(Ω)) and the limiting discrete gradient is ∇w.
  • Identification of weak limits: The two piecewise-constant mesh reconstructions converge to the same weak limit because their L2 difference tends to zero as h→0.This permits identification of the weak limits of both primal and dual components.

6. Properties of discrete solutions

This section derives uniform estimates and entropy inequalities for discrete solutions, establishes solvability of the discrete schemes, and develops compactness tools for their convergence analysis.

  • A priori estimates: Uniform a priori estimates control discrete solutions and their gradients independently of the mesh size under uniformly bounded mesh regularity.The estimates include L∞ bounds and bounds associated with the coercive diffusion operator.
  • Entropy and energy estimates: Discrete entropy and energy inequalities provide nonnegative dissipation terms and control the discrete gradients through coercivity.The estimates are obtained by testing the discrete equations with functions involving A(u) and u.
  • Temporal compactness: Time-translate estimates are derived uniformly in the discretization parameters, supplying the temporal compactness needed in the convergence argument.The estimates are obtained by summing discrete equations over time windows and controlling the resulting terms.
  • Existence of discrete solutions: The discrete problem has at least one solution for every double mesh and time step, obtained through regularization and a Brouwer fixed-point argument.A strictly increasing approximation of A is used before passing to the general nondecreasing case.
  • Scope of the discrete analysis: Uniqueness and continuous dependence of discrete solutions can be established, but the paper treats them as secondary because convergence and continuous well-posedness are already available.This is presented as a scope choice rather than as a failure of the discrete theory.
  • Discrete entropy inequalities: The schemes satisfy discrete entropy inequalities for nondecreasing test functions, with penalization contributions vanishing as h→0.These inequalities include specialized choices that recover estimates needed for compactness and passage to the limit.
  • Flux control: The discrete framework also yields a convex control function for numerical flux differences under the stated continuity and growth assumptions.The function Π_M is continuous, strictly increasing, convex, and satisfies Π_M(0)=Π′_M(0)=0.

7. Convergence and statement of main result

The paper proves convergence of discrete duality finite volume solutions toward the unique entropy solution as mesh size and time step vanish. The proof combines uniform estimates, compactness, discrete duality, Minty–Browder arguments, and strong convergence of reconstructed gradients.

  • Main convergence result: Theorem 7.1 establishes existence, uniform boundedness, and convergence of discrete solutions to the unique entropy solution as h=max{size(T),∆t}→0.The convergence theorem applies to uniformly regular double meshes and associated time steps.
  • Compactness framework: Uniform estimates control the discrete solutions, time and space translates, penalization terms, and reconstructed gradients.These estimates provide the compactness needed for passage to the limit.
  • Identification of limits: The two discrete approximations of A(u) converge to the same limit through the penalization and discrete compactness arguments.The common limit is denoted w, with A(µ)=A(µ*)=w.
  • Identification of limits: Minty–Browder arguments and strict monotonicity identify the nonlinear diffusion limit and yield strong convergence of the discrete gradients in Lp(Q).Weak convergence of the nonlinear flux is upgraded using monotonicity and the limiting variational inequality.
  • Entropy limit: Passing to the limit in the weak and entropy formulations produces an entropy double-process solution, which reduces to the unique entropy solution.Independence from the auxiliary parameters implies strong convergence in Ls(Q) for every finite s.

8. On the choice of FV scheme and various generalizations

The paper compares alternative finite volume approaches and delineates the geometric and modeling scope of conformal DDFV schemes. Their convergence proof relies on mesh restrictions that provide discrete structural properties but limit refinement flexibility and some generalizations.

  • Scheme choices: DDFV schemes are motivated by their convenient treatment of nonlinear diffusion and by discrete integration-by-parts properties.The paper situates DDFV among finite volume, finite element, and related schemes.
  • Scheme choices: Cartesian alternatives can simplify finite volume discretization and avoid entropy double-process solutions, but the 2D schemes are restricted to Cartesian geometries.Their extension to 3D is described as straightforward.
  • Scheme choices: The presented conformal DDFV schemes use double meshes and compare with methods using additional flux or edge unknowns and penalized finite differences.Other schemes may use fewer unknowns or allow broader mesh geometries.
  • Geometric restrictions: The schemes are constrained by orthogonality, Delaunay, and interface conditions, so simple local refinement procedures are unavailable.Within these restrictions, polygonal and polyhedral domains can still be partitioned into suitable triangles and tetrahedra.
  • Geometric restrictions: Non-conformal meshes can preserve discrete duality, but the discrete Poincaré inequality may fail and undermine convergence analysis.The paper identifies conformity as essential for several DDFV convergence properties.
  • Generalizations: The presentation assumes homogeneous Dirichlet data, while inhomogeneous boundary conditions are possible but technically involved.The paper also notes challenges for anisotropic diffusion and higher-dimensional reconstruction.

Appendix A: Proof of uniqueness

Appendix A proves uniqueness by comparing entropy double-process solutions through a Kruzhkov doubling-of-variables argument. Interior and boundary entropy inequalities, initial traces, and monotonicity establish nonnegativity of the comparison distribution and collapse the process variables.

  • Comparison argument: The uniqueness proof adapts Kruzhkov’s doubling-of-variables method to entropy double-process solutions.The comparison uses separate space-time variables and auxiliary process parameters for the two solutions.
  • Comparison argument: A comparison distribution is decomposed into positive and negative parts using algebraic sign identities and monotonicity of the diffusion operator.The decomposition organizes the entropy inequalities for positive and negative solution components.
  • Uniqueness conclusion: The comparison distribution is nonnegative for interior and boundary test functions, becoming a locally finite measure before the boundary argument is completed.The boundary extension uses modified entropy tests and the same comparison framework.
  • Degenerate diffusion: The proof treats separately the regions where the diffusion degenerates and where it is nondegenerate, using tailored entropy tests and diffusion fluxes.The sets are defined by whether the transformed variable lies in the degeneracy set E.
  • Comparison argument: Space-time mollifiers reduce the two-variable comparison inequality to the desired distributional comparison inequality.The test function combines a nonnegative test function with symmetric spatial and temporal mollifiers.
  • Initial trace: The strong initial trace property controls the temporal boundary terms arising in the comparison argument.The initial datum is recovered in the required strong sense for entropy double-process solutions.
  • Uniqueness conclusion: The final comparison forces both process variables to coincide with one another and with the comparison solution, independently of the process parameters.This establishes uniqueness of the entropy solution.

Appendix B: The reconstruction property

Appendix B establishes the geometric reconstruction identity underlying the two-dimensional DDFV construction. It also explains why a direct analogue generally fails for higher-dimensional simplexes.

  • Two-dimensional reconstruction: The reconstruction lemma concerns a triangle, its circumcenter, orthogonal projections, and signed subtriangle areas.These geometric quantities define the affine reconstruction used by the scheme.
  • Higher-dimensional limitation: A proposed multi-dimensional generalization using projections onto face-containing hyperplanes fails except for very particular simplexes.The failure follows from the dimensional mismatch and the analogue of the underlying vector identity.
  • Two-dimensional reconstruction: For two-dimensional polygons admitting a circumscribed circle, the same property can also be proved using the sine theorem.This extends the geometric observation beyond triangles in 2D.
  • Two-dimensional reconstruction: The proof rewrites the geometric identity through projections onto edge directions and their orthogonal complements.The projection decomposition reduces the identity to a vector relation in R2.
  • Two-dimensional reconstruction: The identity is verified using the circumcenter geometry and the Thales theorem, first for a finite set of coefficients and then by linearity for all coefficients.This provides the algebraic consistency needed by the reconstruction formula.
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