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A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

Philippe G. LeFloch, Mai Duc Thanh

arXiv:1105.3074v2math.APcs.NAmath.NA

TL;DR

The paper addresses how solutions to the shallow water Riemann problem behave with variable and possibly discontinuous topography. It constructs a Riemann solver and a well-balanced Godunov-type scheme, reports convergence across most tested regimes, and identifies data exactly on resonance hypersurfaces as a boundary case.

  • Problem

    The paper seeks a complete account of existence and uniqueness or multiplicity for shallow water Riemann solutions with variable topography, information needed to construct a Riemann solver.

    Source: §1 · p. 2

  • Method

    The authors construct an explicit Riemann-solver algorithm and use it to build a well-balanced, quasi-conservative Godunov-type scheme.

    Source: §1 · p. 2, §4.1 · p. 21

  • Results

    The Riemann solver distinguishes existence, uniqueness, and multiplicity; tests show convergence in strictly hyperbolic regions and on both sides of resonance, but not for data precisely on the resonance curve.

    Source: §1 · p. 2, §7 · p. 36

  • Takeaways & Limitations

    The scheme is well-balanced and captures stationary waves exactly.

    Source: §1 · p. 2

    For data precisely on the resonance curve, the scheme showed no convergence; the authors attribute this to a limitation of the physical model, which they say requires further physics in this regime.

    Source: §7 · pp. 36–37

Abstract

from arXiv · show

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.

1. Introduction

The paper develops a Riemann solver and Godunov-type scheme for shallow water equations with variable topography. It characterizes when Riemann solutions exist uniquely or multiply, and reports convergence except for data precisely on resonance hypersurfaces.

1. Introduction

  • A Godunov scheme requires information about the existence and properties of all Riemann solutions because it is based on an exact or approximate Riemann solver.The Riemann data are piecewise constant states on either side of an interface.Source: §1 · p. 2
  • The Riemann problem has a unique solution over a large data domain, while necessary and sufficient conditions identify regions with up to three solutions.The paper presents this as a complete description of the solver, improving on the earlier investigation in LeFloch and Thanh.Source: §1 · p. 2
  • The cited continuation-based Godunov method covers a regime with separated wave speeds, but cannot construct the new wave curves arising in cross-regions.In non-flat-bottom cases, repeated waves can make a solution contain more waves than characteristic families.Source: §1 · p. 2
  • The authors give an explicit algorithm for constructing a Riemann solver in resonant and non-resonant regimes, then use it to design a Godunov-type scheme.The scheme is well-balanced and captures stationary waves exactly.Source: §1 · p. 2
  • Numerical tests show convergence in strictly hyperbolic regions and in resonance regions, except when the scheme takes values precisely on resonance hypersurfaces.In resonance tests, convergence to either the selected solution or another available solution was observed.Source: §1 · pp. 2–3

2. Shallow water equations

This section describes the wave curves and stationary contacts for shallow water flow with variable topography. It establishes when stationary-contact heights exist and how a monotonicity criterion selects admissible contacts.

2.1. Wave curves

  • The system is hyperbolic but not strictly hyperbolic: characteristic speeds coincide on resonance hypersurfaces that divide phase space into three strictly hyperbolic subdomains.The first and second characteristic fields are genuinely nonlinear, while the third is linearly degenerate.Source: §2.1 · p. 3
  • The first two wave families comprise shock and rarefaction curves, while the third describes stationary contacts; shock speeds can coincide with stationary-contact speed.The projected first-family curves are strictly convex and decreasing, whereas the second-family curves are strictly concave and decreasing.Source: §2.1 · pp. 4–5
  • Across a discontinuity, the bottom height stays constant unless the discontinuity is a stationary wave; only stationary waves can change the bottom component of a Riemann solution.With constant bottom, the system reduces to the standard shallow water equations; stationary contacts satisfy jump relations.Source: §2.1 · pp. 4–5

2.2. Properties of stationary contacts

  • A stationary contact to a changed bottom level exists only when condition (2.16) holds; then the nonlinear equation has two roots, distinct when the bottom level is below a_max(U0).These roots give two possible stationary-contact states, with their phase-domain locations determined by the sign of u0 and the root branch.Source: §2.2 · pp. 6–7
  • The Monotonicity Criterion requires the bottom level to vary monotonically along a stationary curve and limits its total variation to |aL − aR|.It selects contacts that do not cross the boundary of strict hyperbolicity: states in G1 or G3 use h1(a), while states in G2 use h2(a).Source: §2.2 · pp. 7–8
  • The paper computes h1(a) with regula falsi on [0, hmin] and h2(a) with Newton’s method from a starting point larger than h*.The two roots lie on opposite sides of h*, and are distinct when the root-existence inequality is strict.Source: §2.2 · pp. 6, 8
  • Some physical applications may require a different jump relation for the nonconservative product, particularly one allowing energy dissipation; the paper does not address that issue further.Source: §2.2 · p. 8

3. The Riemann problem revisited

This section constructs explicit Riemann solutions in coinciding- and opposite-sign eigenvalue regimes and identifies where solutions exist uniquely or multiply. In Regime (B), the problem may have one or three solutions depending on the topography and wave-curve configuration, with the solution set depending continuously on the Riemann data.

3. The Riemann problem revisited

  • The section improves on by combining earlier constructions to enlarge existence domains and precisely determine where solutions are unique or multiple.By the transformation x 7→−x and u 7→−u, constructions around C− follow from those around C+.Source: §3 · p. 9

3.1. Regime (A). Eigenvalues at UL with coinciding signs

  • Construction (A1) has necessary and sufficient existence conditions and covers a large neighborhood crossing the strictly hyperbolic boundary; (A2) admits three discontinuities with the same zero speed.Construction (A2) also extends the solution domain to a region in G2 and G1 far from UL.Source: §3.1 · pp. 10–11
  • In Regime (A), the Riemann problem has at most one solution under the stated geometric conditions, but can have three when the relevant curves meet in the specified configuration.The theorem distinguishes uniqueness from multiplicity by the locations of the states and wave curves.Source: §3.1 · pp. 12–13

3.2. Regime (B). Eigenvalues at UL with opposite signs

  • In Regime (B), for a higher-positioned right state and aL>aR, a solution can proceed through a 1-rarefaction to C+, a stationary jump, then a 1-wave and a 2-wave.The left state lies in the closure of G2, with λ1(UL) < 0 = λ3(UL) < λ2(UL); the passage notes two solution types depending on whether aL ≥ aR.Source: §3.2 · p. 14
  • Construction (B2), applicable only when aL > aR, has three waves whose speeds coincide and all equal zero; all solution states can lie within an arbitrarily small ball centered on C+.The construction proceeds from bottom height aL to a in [aR, aL], then by a zero-speed 1-shock from M to N, followed by a stationary wave from N to P that shifts the a-component to aR.Source: §3.2 · p. 15
  • For aL ≤ aR, the Riemann problem has a unique solution when Ko lies below or on WB2(UR).The construction argument identifies a unique intersection of WB2(UR) with the combined curve W1(U2) ∪ Wa1(UL) under this condition.Source: §3.2 · p. 18
  • When aL > aR, Uo1 above U#2 on W3(U1) yields three distinct solutions; otherwise, there is at most one.The theorem’s multiple-solution condition also describes Uo1 above WB2(UR) and U#2 below that curve.Source: §3.2 · pp. 18–19
  • The reported numerical experiments, illustrating the cases of Theorem 3.2 in Tables B1, B2, and B3, all give the same result: ho1 > h#2.Source: §3.2 · p. 19

3.3. Remark on the continuous dependence of solutions

  • Although some data admit up to three solutions, the set of solutions depends continuously on the Riemann data across constructions and as UL crosses C+.As UL tends to C+ from either side, the corresponding constructions approach one another; for aL ≥ aR, the solutions eventually coincide on C+.Source: §3.3 · p. 20

4. A Godunov-type algorithm

The scheme advances cell averages using local Riemann solutions and a solver-selection algorithm. Its construction is quasi-conservative and exactly preserves stationary equilibria.

4.1. A well-balanced quasi-conservative scheme

  • The cell update uses local Riemann solutions to define an intermediate function, then sets each new cell value to its mean over that cell.A CFL condition keeps consecutive local Riemann solutions from coinciding.Source: §4.1 · pp. 20–21
  • Because the topography component is constant within each cell, the source term vanishes there and the Godunov scheme is quasi-conservative.For equilibrium initial data, the exact local solutions are constant, so the scheme captures those equilibria exactly.Source: §4.1 · p. 21

4.2. Numerical Riemann solver

  • The numerical Riemann solver computes interface states using constructions for the relevant wave patterns, with alternative constructions yielding the same interface states in one case.The method selects among available solvers by preferring (A1) or (B3), producing a deterministic Godunov algorithm.Source: §4.2 · pp. 21, 24, §4.3 · p. 24

5. Numerical experiments (I). The non-resonant regime

The non-resonant tests compare Godunov approximations with exact Riemann solutions across mesh sizes. The reported tests show exact capture of a stationary contact and closer approximations as meshes are refined.

5.1. Test 1

  • The stationary contact wave is captured exactly by the Godunov method using the exact Riemann solver.The test reports 500 mesh points at t = 0.1.Source: §5.1 · p. 25, Figure 13 · p. 26

5.2. Test 2

5. Numerical experiments (I). The non-resonant regime

  • Across the tests presented, approximate solutions converge to the exact solution as mesh size decreases.The reported mesh refinements use 500, 1000, and 2000 points.Source: §5 · p. 25, §5.3 · p. 28

5.3. Test 3

  • In Test 3, the approximate solution becomes closer to the exact solution as mesh size decreases.The solution has a 1-shock, a stationary contact, and a 2-shock.Source: §5.3 · p. 28, Figure 16 · p. 28

6. Numerical experiments (II). The resonance regime

The tests examine Riemann data in resonance regions and show that convergence depends on the data and, when multiple solutions exist, the preferred solver.

6.1. Test 4

6.2. Test 5

6.3. Test 6

  • With three distinct Riemann solutions, Solver (A1) and Solver (A2) yield approximate solutions that approach the exact solution, but Solver (A3) does not.The three solution structures include a stationary solution; the reported convergence depends on which available solver is preferred.Source: Figure 22 · p. 32, Figure 23 · p. 32, §6.3 · p. 33

6.4. Test 7

7. Concluding remarks

The concluding remarks report exact equilibrium capture and convergence in strictly hyperbolic and some resonant cases, but not for data precisely on the resonance curve. The paper characterizes the Riemann solutions and builds a well-balanced, quasi-conservative Godunov scheme around a numerical Riemann solver.

7. Concluding remarks

  • The paper characterizes existence, uniqueness, and multiplicity of shallow-water Riemann solutions, computes possible solutions, and defines a well-balanced, quasi-conservative Godunov scheme.The scheme is tested across several regimes.Source: §7 · p. 36
  • The proposed scheme captures equilibrium states exactly.Source: §7 · p. 36
  • In strictly hyperbolic phase-space domains, the scheme converges to the uniquely defined Riemann solution.Source: §7 · p. 36
  • In resonance regions, tests show convergence both to the selected solution and to another available solution, consistent with the availability of multiple solutions.The paper notes that up to three solutions may be available.Source: §7 · p. 36
  • For states precisely on the resonance curve, the scheme shows no convergence; the authors attribute this behavior to a limitation of the physical model.They suggest that further physics, potentially including higher-order effects or a kinetic relation, may be needed in this regime.Source: §7 · pp. 36–37
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