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An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry

Niklas Wintermeyer, Andrew R. Winters, Gregor J. Gassner, David A. Kopriva

arXiv:1509.07096v2math.NAcs.NA

TL;DR

The paper addresses the challenge of constructing robust, accurate shallow-water methods that preserve steady states over non-constant and discontinuous bathymetry. It develops a high-order DGSEM using equivalent flux differencing, and reports a conservative, entropy-stable, well-balanced approximation for unstructured curved quadrilateral meshes.

  • Problem

    Shallow-water discretizations must remain stable and accurate while preserving steady states, a particular challenge with discontinuous bottom topography where special source-term treatments are needed.

    Source: §1 · pp. 1–2

  • Method

    The authors reformulate the DGSEM as an equivalent subcell flux-differencing scheme and select volume, interface, and source-term discretizations to establish entropy and well-balanced properties.

    Source: §3 · p. 4, §4.1 · p. 11

  • Results

    The method is a high-order entropy-stable and well-balanced DGSEM for discontinuous bathymetry on unstructured curved quadrilateral grids, with discrete conservation of total energy.

    Source: §1 · p. 2, §4.1 · p. 12, §4.2 · p. 15

  • Takeaways & Limitations

    Placing element boundaries at bathymetry discontinuities allows the method to model realistic bottom topographies, and six numerical examples demonstrate the theoretical findings.

    Source: §1 · p. 2, §6 · p. 23

    The entropy-stable approximation can capture shock and rarefaction waves but is not designed to eliminate overshoots, so additional DG shock-capturing techniques are needed to remove remaining oscillations.

    Source: §5.6.2 · p. 22

Abstract

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We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

1. Introduction

The introduction motivates robust, well-balanced shallow-water numerics for non-constant and discontinuous bathymetry, then presents a high-order DGSEM that is entropy stable on unstructured curved quadrilateral grids. Its flux-differencing formulation supports conservation and the paper’s entropy and well-balanced results.

1. Introduction

  • The shallow-water equations arise by simplifying the Euler equations under hydrostatic balance, reflecting flows whose vertical scales are much smaller than their horizontal scales.For non-constant bottom topography, the equations take the form of a hyperbolic system of balance laws.Source: §1 · p. 1
  • Discontinuous bottom topography makes steady-state preservation difficult and requires special source-term discretizations.Failure to preserve the lake-at-rest state can generate mesh-scale spurious waves, or a “numerical storm”.Source: §1 · pp. 1–2
  • The method extends prior one-dimensional shallow-water DGSEM work to multiple dimensions and potentially discontinuous bottom topographies.The paper’s outline describes a conservative and entropy-conservative curvilinear-mesh scheme, a well-balanced source-term treatment, and added interface dissipation for regimes that may contain shocks.Source: §1 · p. 2
  • The proposed high-order DGSEM handles discontinuous bathymetry on unstructured curved quadrilateral grids and is entropy stable and well-balanced.Placing element boundaries at bathymetry discontinuities supports modeling realistic topographies; satisfying metric identities is critical to the stated entropy-conservation and well-balanced proofs on arbitrary meshes.Source: §1 · p. 2
  • Reformulating the split-form DGSEM as equivalent flux differencing enables proofs of conservation and entropy stability while showing how existing code can incorporate entropy-stable fluxes.The paper also uses this flexibility to prove the approximation remains well-balanced.Source: §1 · p. 2

2. Shallow water equations

The two-dimensional shallow-water equations describe water height and momentum evolving under fluxes and bathymetry-dependent source terms. Their total energy supplies a mathematical entropy, which is conserved for smooth solutions and must satisfy an inequality when discontinuities occur; the lake-at-rest state motivates well-balanced discretization.

2. Shallow water equations

  • The state vector contains water height and its two momentum components, with fluxes in both spatial directions and a source term determined by bottom-topography gradients.The text identifies h as water height above bottom topography b, total height as H = h + b, and g as gravitational acceleration.Source: §2 · p. 3
  • The convex mathematical entropy for the shallow-water system is total energy, conserved for smooth solutions and required to decay for discontinuous solutions.The paper calls a numerical approximation entropy stable when it discretely satisfies the entropy condition.Source: §2 · p. 3
  • The entropy fluxes are determined using compatibility relations between entropy variables and physical fluxes, then used to derive the entropy conservation law.The derivation pre-multiplies the balance law by the entropy variables and applies the stated compatibility conditions.Source: §2 · p. 3
  • The lake-at-rest state is a steady configuration in which flux and source terms balance, and preserving it discretely defines a well-balanced method.Without well-balancedness, spurious waves on the order of the mesh-size truncation error can pollute the approximation.Source: §2 · p. 3

3. Nodal discontinuous Galerkin spectral element method

The section develops a nodal DG spectral element method on curvilinear quadrilateral grids, linking its summation-by-parts structure to an equivalent subcell flux-differencing formulation. This formulation supports split-form choices for conservative, entropy-conservative, and well-balanced approximations.

3.1. Conservation law in curvilinear coordinates

  • Curvilinear quadrilateral elements are mapped to a reference square, where the conservation law is expressed using a Jacobian and contravariant fluxes.Source: §3.1 · p. 4

3.2. Polynomial approximation on the reference element

  • The method uses degree-N nodal polynomials at Legendre-Gauss-Lobatto points; isoparametric boundary curves of order at most N with linear blending guarantee free-stream preservation.Free-stream preservation requires the metric identities to hold discretely.Source: §3.2 · pp. 5–6

3.3. Discontinuous Galerkin spectral element method (DGSEM)

  • The DGSEM's weak and strong forms are algebraically equivalent because Legendre-Gauss-Lobatto quadrature satisfies a summation-by-parts rule, while numerical interface fluxes couple neighboring elements.Source: §3.3 · p. 7

3.4. An equivalent subcell flux differencing form

  • Diagonal-norm SBP operators yield an equivalent subcell flux-differencing form, generalized to curvilinear metrics, that supports conservative split forms and entropy-conservative DGSEM choices.Two-point entropy-conserving fluxes retain high order and entropy conservation; the telescoping form supports conservation while retaining split-form stabilization.Source: §3 · p. 4, §3.4 · pp. 9–10
  • Products of arithmetic means reproduce discrete split forms for quadratic and triple products, including the form proposed by Kennedy and Gruber.Source: §3.4 · p. 10
  • Flux differencing permits alternative PDE split forms for entropy-conservative approximations; changing a standard DGSEM’s volume contributions and selecting suitable fluxes makes it entropy stable.Source: §3.4 · p. 10
  • Symmetric volume fluxes preserve high-order accuracy and conservation; carefully chosen volume and surface fluxes yield entropy conservation, and independent flux choices enable a well-balanced approximation.Source: §3.4 · p. 11

4. Entropy stable DGSEM for the 2D shallow water equations

The section develops entropy-conserving and entropy-stable DGSEM formulations for the shallow water equations on curvilinear meshes. Their flux-differencing and bathymetry-source discretizations yield stated conservation, entropy, and well-balanced properties.

4.1. ECDGSEM on curvilinear meshes

  • The flux-differencing form permits established conservation and entropy-conservation results to apply while allowing distinct volume, surface, and source-term discretizations.The source term is split into volume and surface contributions, with interface averages and jumps.Source: §4.1 · p. 11
  • The curvilinear ECDGSEM discretely conserves mass and momentum for constant bottom topography, conserves total energy as entropy, and is well-balanced for arbitrary bottom topographies.Its conservation follows from flux differencing, while entropy conservation uses entropy-conserving volume and surface fluxes.Source: §4.1 · pp. 11–12
  • The well-balanced source-term construction approximates bottom-topography derivatives using split-form volume contributions and interface contributions based on jumps.The source discretization uses metric identities to cancel extraneous terms and obtain a more computationally efficient volume form.Source: §4.1 · pp. 12–13

4.2. Entropy stable DGSEM

  • The curvilinear ESDGSEM discretely conserves mass and momentum for constant bottom topography, is entropy stable, and is well-balanced for arbitrary bottom topographies.The formulation combines entropy-stable interface fluxes with the specified source-term discretization.Source: §4.2 · p. 15
  • The ESDGSEM replaces entropy-conserving interface fluxes with entropy-stable fluxes whose dissipation depends on jumps in entropy variables.The dissipation is negative in the discrete entropy equation; it acts for large jumps such as shocks and is nearly absent for small jumps in well-resolved smooth solutions.Source: §4.2 · p. 15

5. Numerical results

The numerical tests verify the schemes’ convergence and conservation properties, including on discontinuous bathymetry. They also exercise the ESDGSEM in a partial dam-break simulation.

5. Numerical results

  • The test suite includes convergence, conservation, and well-balancedness checks, bore-shear and potential-vorticity tests, and a partial break of a parabolic dam with discontinuous downstream topography.The partial dam-break simulation exercises each component of the ESDGSEM approximation.Source: §5 · p. 15

5.1. Convergence

  • Both schemes are spectrally accurate for smooth problems, with exponential convergence until time-integration errors dominate; ECDGSEM convergence is suboptimal at odd polynomial degrees.The reported limits are N = 16 for ECDGSEM and N = 15 for ESDGSEM at ∆t = 1/2000, and N = 17 and N = 16, respectively, at ∆t = 1/4000.Source: §5.1 · p. 16, Figure 2 · p. 17, Figure 3 · p. 17

5.2. Conservation of mass and momentum

  • The tests examine mass, momentum, and energy conservation, including mass and entropy conservation for non-constant, discontinuous bottom topography.For non-constant topography, the momentum equations are balance laws rather than conservation laws.Source: §5.2 · p. 16

5.2.1. Dam break over a flat bottom

  • On a flat bottom, mass and momentum are conserved to machine precision, while total-energy error converges to machine precision at the time integrator’s fourth order as the time step shrinks.The energy error reflects dissipation from the time integrator.Source: §5.2.1 · p. 16, Table 1 · p. 18

5.2.2. Dam break over a discontinuous bump

  • For the discontinuous-bottom test, mass is conserved to machine precision and total-energy error decreases at fourth order as the time step is refined.The test uses a bottom topography restricted to one element.Source: §5.2.2 · p. 18, Table 2 · p. 18

5.3. Well-balancedness over a discontinuous bottom

The tests show round-off-level preservation of a lake-at-rest state over discontinuous bathymetry, and demonstrate how entropy-stable dissipation changes shock and wave behavior. The examples also examine a discontinuous bump and nonlinear wave breaking.

5.3. Well-balancedness over a discontinuous bottom

  • The L2-error in total water height is on the order of round-off for both ECDGSEM and ESDGSEM in the discontinuous-bottom lake-at-rest test.The test varies polynomial degree on a curvilinear mesh.Source: §5.3 · p. 18, Table 3 · p. 18

3.5 −b3(x, y)

  • For the initial state, the passage specifies that for x > 5, both velocity components u(x, y, 0) and v(x, y, 0) are zero.Source: §3.5 · p. 19

2.5 −b3(x, y)

  • The bump is a smooth-topped box centered at (5.0, 5.0), with side lengths of 2 and discontinuities aligned with cell interfaces.Source: §2.5 · p. 19

5.4. Dam break over a discontinuous bump

  • For the dam break over a discontinuous bump, ECDGSEM produces severe post-shock ringing, while ESDGSEM smooths it except near the shock front.The entropy-stable approximation is not completely oscillation-free.Source: §2.5 · p. 19, Figure 4 · p. 19

5.5. Non-linear breaking shallow waters waves

  • In the nonlinear wave test, higher-amplitude gravity waves begin breaking at t ≈ 0.5, forming peaks near the walls and bores that move in the negative x-direction.Source: §5.5 · p. 21, Figure 6 · p. 21
  • At N = 3 and N = 7, the computations show more resolution in smooth flow regions but oscillations near discontinuities; generated potential vorticity follows the discontinuities’ shape.The authors attribute potential-vorticity noise at element boundaries, especially for N = 7, to dissipation that ensures entropy stability but does not guarantee an overshoot-free solution.Source: §5.5 · pp. 20–21

5.6. Parabolic dam break

The parabolic-dam tests verify well-balancedness on a curvilinear mesh with discontinuous bottom topography and demonstrate a partial-break simulation. The entropy-stable method captures the resulting waves, but its dissipation does not eliminate all oscillations.

5.6. Parabolic dam break

  • The test sequence first checks well-balancedness before dam failure, using two lake-at-rest regions and discontinuous bottom topography downstream.Source: §5.6 · p. 21

5.6.1. ESDGSEM well-balancedness

  • For the lake-at-rest test on either side of the parabolic dam, total-water-height error is on the order of machine precision, including on the side with discontinuous bottom topography.The test uses a curvilinear mesh and reports errors at T = 5 for various polynomial degrees.Source: §5.6.1 · p. 22, Table 4 · p. 22

5.6.2. Partial dam break with discontinuous bottom topography

  • The partial break assumes instantaneous failure only for y ∈ [−0.5, 0.5], where the two states interact.Source: §5.6.2 · p. 22
  • The p-refinement study finds the partial-break waves well resolved at N = 5 and N = 7, and the N = 5 simulation shows eddies near the dam break.The simulation combines a discontinuous solution, a curvilinear mesh, and discontinuous bottom topography.Source: §5.6.2 · p. 22
  • The partial-dam-break solution captures shock and rarefaction waves, while the standard DGSEM is unstable on this problem even with very small time steps.Source: §5.6.2 · p. 22

6. Conclusion

The conclusion describes a high-order entropy-conserving and entropy-stable DGSEM for shallow water equations on general curvilinear meshes. It explains the method’s conservation, entropy, and well-balanced properties, alongside the trade-off between shock dissipation and residual oscillations.

6. Conclusion

  • An equivalent flux-differencing reformulation supports local conservation and, with chosen volume and surface fluxes, an entropy-conserving scheme on general curvilinear meshes.Source: §6 · p. 23
  • A particular nonlinear source-term discretization preserves well-balancedness for discontinuous bottom topographies in general curvilinear coordinates.Source: §6 · p. 23
  • The entropy-conserving scheme can have significant post-shock oscillations; added entropy-stable dissipation controls overshoots but is not designed to make the method overshoot-free.Six numerical examples demonstrate the theoretical findings, and the partial parabolic-dam break exercises each component of the approximation.Source: §6 · p. 23

Appendix A. Proof of Prop. 1

The proof derives the flux-difference formula by working through component differences, using SBP matrix structure and flux symmetry, then extending the argument across directions.

Appendix A. Proof of Prop. 1

  • The proof starts with a single flux difference for components i = 1 and i = 2 at fixed j, then expands and subtracts component expressions to cancel terms.Source: Appendix A · p. 23
  • The argument uses the nearly skew-symmetric SBP matrix and symmetry of the arithmetic mean and volume flux to collect the terms into one sum.Source: Appendix A · p. 23
  • It generalizes the flux-difference calculation across i and the y-direction, then premultiplies by M^-1 to obtain the flux-differencing result.Source: Appendix A · p. 24

Appendix B.1. Proof of Property 1.2

The proof establishes that the chosen volume fluxes satisfy the discrete entropy-conservation condition, alongside entropy-conservative interface fluxes.

Appendix B.1. Proof of Property 1.2

  • The proof starts from entropy-conservative interface fluxes and verifies entropy conservation for the selected volume fluxes.The displayed volume flux components are the fluxes used in that verification.Source: Appendix B.1 · pp. 24–25
  • The discrete criterion relates entropy-variable jumps contracted with volume fluxes to entropy-flux jumps and the auxiliary source contribution.The source discretisation contributes terms involving the water velocity and bathymetry jumps.Source: Appendix B.1 · p. 24
  • The volume fluxes are shown to be entropy conserving in the x-direction, with the y-direction handled analogously.The proof concludes that both volume fluxes satisfy the property.Source: Appendix B.1 · p. 25

Appendix B.2. Proof of Property 1.3

The proof verifies well-balancedness by showing that the lake-at-rest state is preserved, including across discontinuous bathymetry interfaces.

Appendix B.2. Proof of Property 1.3

  • For lake-at-rest data, h + b = const and u = v = 0, the discrete time derivatives vanish, so the scheme is well-balanced.The continuity equation vanishes directly; the momentum equations are checked in the proof.Source: Appendix B.2 · pp. 25, 27
  • The source-term discretisation uses a quadratic split form and adds an interface contribution for discontinuous bottom topography.That additional interface term vanishes when the bottom topography is continuous.Source: Appendix B.2 · p. 26
  • In the volume momentum terms, exact differentiation of constant h + b and the metric identities make the remaining contributions cancel.The proof uses D(h + b) = 0 for constant total water height.Source: Appendix B.2 · pp. 26–27
  • The interface contributions also cancel when h + b remains constant across interfaces, completing the momentum balance.The proof treats the hv equation by an analogous argument.Source: Appendix B.2 · p. 27

Appendix C. Algorithmic Description of the ESDGSEM

The appendix describes how to adapt a standard DGSEM time-derivative routine into an efficient flux-differencing implementation of the ESDGSEM, including source-term contributions for curved, unstructured elements.

Appendix C. Algorithmic Description of the ESDGSEM

  • The algorithm discussion focuses on a single spectral element and states that it extends to standard DGSEM implementations.The example implementation follows a mapped DGSEM routine for curved quadrilateral elements.Source: Appendix C · p. 27
  • The source-term computation is split into volume and surface parts, with edge jumps in bottom topography handled using the adjacent elements’ local edge information.The surface contribution is added to the edge-flux routine for convenience on the unstructured mesh.Source: Appendix C · p. 28
  • Bottom derivatives and source contributions are precomputed or stored, including volume terms, edge jumps, and average water heights.The bottom topography is time-independent, allowing its derivative-related quantities to be precomputed.Source: Appendix C · pp. 27–29
  • High-order x- and y-flux-difference routines reformulate the volume contributions using solution and metric data on a single element.The global time derivative is assembled by looping over mesh elements.Source: Appendix C · pp. 27–30
  • The implementation replaces the standard DG derivative with flux differencing, entropy-stable interface fluxes, and volume and surface source contributions.The authors describe this as a restructuring of a standard DGSEM time-derivative routine.Source: Appendix C · p. 29

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