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Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations

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

arXiv:1604.06618v3math.NA

TL;DR

Under-resolved high-order DG methods for compressible Euler equations can become unstable, while exact quadrature-based stability arguments are impractical for their rational fluxes. The paper uses SBP-based volume flux differencing to systematically generate quadratic and cubic split forms, evaluates their kinetic-energy preservation, and finds improved robustness in three-dimensional Taylor–Green vortex simulations, including cases where over-integrated DG crashes.

  • Problem

    Under-resolved high-order DG approximations for compressible Euler equations can suffer nonlinear instability, while exact integration is impractical for their rational flux nonlinearities.

  • Method

    The paper uses SBP volume flux differencing and selected symmetric, consistent subcell fluxes to construct a unified DG framework for quadratic and cubic split forms.

  • Results

    The new split-form DG schemes are more robust than standard DG with over-integration in tested turbulent vortex-dominated cases, while kinetic-energy-preserving fluxes retain high-order accuracy and kinetic-energy preservation.

  • Takeaways & Limitations

    The flux-form dictionary supports systematic construction of established and new split forms, with some variants offering computational advantages over complex entropy-stable alternatives.

  • Takeaways & Limitations

    The entropy-variable stability strategy requires exact variational-term evaluation, which is impossible or impractical for compressible Euler fluxes with fixed standard quadrature.

Abstract

from arXiv · show

Fisher and Carpenter (\textit{High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518--557, 2013}) found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we show we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid Taylor-Green vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

1. Introduction

The paper addresses instability in under-resolved high-order DG approximations for compressible Euler flows and develops split-form alternatives using the SBP framework. It connects subcell flux choices with quadratic and cubic split forms, including established formulations with kinetic-energy-preserving properties.

  • Stabilisation motivation: Under-resolved high-order DG discretisations can suffer aliasing-driven nonlinear instability and crash without additional stabilisation.High-order schemes have lower inherent numerical dissipation, making under-resolved turbulence and shocks particularly challenging.
  • Stabilisation strategies: Polynomial de-aliasing improves scalar nonlinear stability, but the corresponding estimate does not extend to compressible Euler systems.For Euler fluxes, rational dependence on conserved or entropy variables prevents exact integration with a fixed standard quadrature rule.
  • SBP approach: Entropy-stable SBP-DG constructions avoid exact inner-product evaluation by using a two-point numerical flux that is entropy conservative in a finite-volume discretisation.The result applies to diagonal-norm SBP operators, including the collocated DGSEM.
  • Split formulations: Alternative split formulations rewrite nonlinear Euler advection terms in multiple equivalent ways, offering another de-aliasing strategy for high-order methods.The paper focuses on split forms for quadratic and cubic products and their DGSEM discretisation.
  • Split-form framework: The SBP property enables conservative DGSEM discretisations of selected split forms, including formulations associated with Ducros, Kennedy and Gruber, and Morinishi.The paper examines which formulations can produce formally kinetic-energy-preserving discretisations.

2. The Nodal Discontinuous Galerkin Spectral Element Method

The nodal DGSEM uses Gauss-Lobatto collocation, diagonal mass matrices, and SBP differentiation on mapped hexahedral elements. Its strong-form discretisation combines collocated volume fluxes with numerical surface fluxes and explicit Runge–Kutta time integration.

  • Element mapping: Each hexahedral element is mapped to the reference cube, with mapping degree at most N to preserve the free stream.Non-curved elements use a trilinear mapping and Cartesian metric simplifications.
  • Polynomial approximation: Conservative variables are approximated componentwise by degree-N polynomials at the tensor-product Gauss-Lobatto nodes.The Lagrange basis has the cardinal property and yields quadrature weights with precision 2N −1.
  • Nodal basis and SBP operators: The DGSEM uses Gauss-Lobatto nodes for both interpolation and numerical integration, producing a diagonal-norm SBP operator.The mass matrix is diagonal, while Q = MD satisfies Q + Q^T = B.
  • Strong-form discretisation: The strong-form DGSEM approximates each flux equation at nodal points and introduces numerical surface fluxes to resolve interelement jumps.Surface fluxes depend on left and right interface states and are specified separately from the volume discretisation.
  • Flux evaluation and time integration: Standard collocation evaluates nonlinear fluxes from nodal solution values, whereas over-integration requires an L2 projection onto the degree-N polynomial space.The resulting semidiscrete equations are advanced with a low-storage five-stage fourth-order explicit Runge–Kutta method.

3. Split form stabilisation for DGSEM

The framework rewrites DGSEM volume terms through flux differencing, then selects numerical volume fluxes to recover standard, quadratic, and cubic split forms. It also characterizes kinetic-energy preservation and combines compatible volume and surface fluxes for robust, dissipative discretisations.

  • 3.1. DGSEM with numerical volume flux function: Diagonal-norm SBP operators admit an equivalent subcell finite-volume flux-differencing formulation that preserves conservation through telescoping fluxes.The formulation also enables entropy-conserving discretisations without exact integration.
  • 3.1. DGSEM with numerical volume flux function: The volume fluxes F#, G#, and H# are kept general, while their choices are coupled to the numerical surface fluxes to limit possible DGSEM variants.The paper notes that volume and surface fluxes could differ in principle, but restricts the combinations considered.
  • 3.2. Split form DSGEM: Arithmetic means, products of two arithmetic means, and products of three arithmetic means recover the standard derivative, discrete quadratic, and discrete cubic split forms, respectively.This identity applies to generic nodal vector fields and provides the algebraic basis for the split-form construction.
  • 3.2. Split form DSGEM: Theorem 1 shows that translating quadratic and cubic split forms into numerical volume fluxes generates high-order, consistent DGSEM discretisations for Morinishi, Ducros, Kennedy–Gruber, and Pirozzoli formulations.The result provides a unified way to reproduce these alternative Euler operator split forms.
  • 3.3. Kinetic energy preservation of the split forms: Kinetic-energy preservation requires volume fluxes satisfying the stated conditions, allowing advective terms to be written conservatively while pressure work remains non-conservative.This is the Jameson sense of kinetic-energy preservation used in the paper.
  • 3.3. Kinetic energy preservation of the split forms: The resulting kinetic-energy balance is consistent with the continuous balance, so advective terms do not dissipate discrete total kinetic energy.The paper identifies this property as important for low artificial dissipation of kinetic energy.
  • 3.3. Kinetic energy preservation of the split forms: When volume and surface fluxes satisfy the same conditions, the multi-element DGSEM remains kinetic-energy preserving; variants MO, KG, PI, and CH are included.The corollary extends the volume-term result to element interfaces.
  • 3.4. Numerical surface flux functions: LLF-type dissipation guarantees kinetic-energy dissipation in the first four components, while CH’s fifth component additionally guarantees entropy dissipation; IR and CH stabilisations are entropy stable.These properties concern the numerical surface-flux stabilisation terms.

4. Numerical Investigations

The numerical investigations show that interface stabilisation restores optimal convergence and robust high-order simulations, while split forms differ substantially in kinetic-energy preservation but only slightly in entropy dissipation. Except for MO, stabilised split forms improve robustness over over-integration, with cubic forms performing better than quadratic splitting for compressible flow.

  • Convergence: Interface stabilisation restores optimal N + 1 convergence in h for all tested split schemes.Without stabilisation, odd polynomial degrees exhibit an odd/even effect and converge only at order N.
  • Robustness: Without interface stabilisation, the schemes are nearly dissipation-free but fragile, with higher polynomial degree and spatial resolution making successful completion harder.The auxiliary conservation-property study therefore omits stabilisation, while robustness tests use stabilising terms.
  • Entropy conservation and kinetic energy preservation: IR and CH conserve discrete entropy to machine precision, whereas other schemes exhibit only very small entropy decay.The entropy dissipation of the other formulations is described as arguably negligible.
  • Entropy conservation and kinetic energy preservation: KG and PI best preserve total kinetic energy, while other discretisations lose up to 10% and IR performs worst.KG, PI, and CH are constructed so discrete pressure work, rather than advective terms, changes total kinetic energy.
  • Entropy conservation and kinetic energy preservation: Changing CH pressure discretisation to a simple arithmetic mean makes its kinetic-energy evolution resemble KG and PI but removes entropy conservation.The results show that pressure discretisation strongly affects actual kinetic-energy preservation, despite CH’s formal properties.
  • Robustness: All stabilised split schemes except MO remain robust at very high polynomial degrees, even under severe under-resolution.The stable schemes successfully run to the final time in configurations where over-integrated DG discretisations could not.
  • Robustness: At fixed 643 degrees of freedom, kinetic-energy dissipation rates change little across polynomial degrees, although numerical viscosity estimates decrease for higher-order discretisations.The reduced numerical-viscosity estimate is associated with higher enstrophy, while actual kinetic-energy dissipation remains similar.
  • Robustness: Stabilised split forms except MO significantly improve robustness over polynomial de-aliasing with over-integration, and cubic forms are more robust than quadratic splitting for compressible flow.The comparison is based on three-dimensional inviscid Taylor–Green vortex simulations, including a higher-compressibility case at Ma = 0.4.

5. Final Remarks

The paper establishes a unified translation framework between split forms and numerical volume fluxes, enabling systematic construction of new DG schemes. Numerical results show improved robustness for several split forms, while robustness depends on the formulation, Mach number, resolution, and polynomial degree.

  • 5.1. Discussion: The Lemma 1 identities provide a dictionary translating split forms into flux forms and vice versa.The dictionary supports translation of known compressible Euler formulations and construction of new split forms.
  • 5.1. Discussion: Choosing symmetric and consistent flux approximations generates a multitude of new split forms.The Roe-variable example reformulates the flux using only quadratic products and produces a new numerical volume flux.
  • 5.2. Conclusion: The unified volume-flux framework constructs split forms for quadratic and cubic products and yields kinetic-energy-preserving, high-order accurate discretisations.Changing the numerical volume flux generates new DGSEM variants while preserving the stated accuracy and kinetic-energy property.
  • 5.2. Conclusion: The new DG schemes are more robust than DG with over-integration in the reported numerical assessment.Configurations that crash with over-integration can be completed using split-form DGSEM or entropy-stable DGSEM.
  • 5.2. Conclusion: Robustness is formulation-dependent: MO crashes broadly, DU is less robust at higher Mach numbers, and KG and PI offer a reported balance of robustness and computational effort.For N = 3 without interface stabilisation, all proposed split forms except standard and MO run at tested resolutions up to 256^3.

Appendix A.1. Proof of Lem. 1 [Discrete split forms]

The appendix proves discrete split-form identities by expanding products of averages and applying the SBP matrix property that each row sums to zero. These identities connect flux-difference expressions with split-form matrix products.

  • Appendix A.1. Proof of Lem. 1 [Discrete split forms]: The proof uses D = M^-1Q and the vanishing row sums of Q to derive the discrete split-form identities.The final expanded expressions are condensed into matrix-vector form.
  • Appendix A.1. Proof of Lem. 1 [Discrete split forms]: The discrete identities therefore provide the algebraic bridge between SBP flux differences and quadratic or cubic split forms.This conclusion follows from the two-average and triple-average expansions together with D = M^-1Q.
  • Appendix A.1. Proof of Lem. 1 [Discrete split forms]: For products of two averages, expanding Q_im(a_i + a_m)(b_i + b_m) reduces to the corresponding split-form terms because each row of Q sums to zero.After scaling by the i-th term of M^-1, the flux-difference result becomes the i-th row of the split form.
  • Appendix A.1. Proof of Lem. 1 [Discrete split forms]: The same expansion strategy is applied to triple products of averages to obtain the cubic split-form identity.The proof starts from Q_im(a_i + a_m)(b_i + b_m)(c_i + c_m) and uses the same SBP cancellation property.
  • Appendix A.1. Proof of Lem. 1 [Discrete split forms]: The resulting cubic expression is written in matrix-vector form as seven derivative-product terms.The displayed identity combines derivatives of a, b, and c with products of the remaining variables.

Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]

The appendix proves kinetic-energy preservation by inserting the numerical flux structure into the discrete kinetic-energy balance. Symmetry and consistency reduce the advective contributions to consistent, high-order approximations of the continuous kinetic-energy fluxes and pressure work.

  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: The discrete kinetic-energy balance combines the mass and three momentum equations with the numerical fluxes F#,1, G#,1, and H#,1.The velocity averages are defined as arithmetic averages of paired states.
  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: The proof rewrites the time derivative of kinetic energy using the time derivatives of density and momentum.The volume parts of the continuity and momentum equations are then evaluated at a single node.
  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: The pressure-work discretisation is consistent and high-order accurate when the numerical pressure trace is consistent and symmetric.This applies to the discrete approximation of u p_x + v p_y + w p_z.
  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: Summing the advective terms produces symmetric two-state expressions consistent with the continuous kinetic-energy fluxes in all three spatial directions.The x-, y-, and z-direction terms correspond respectively to f = 1/2ρu(u^2 + v^2 + w^2), g = 1/2ρv(u^2 + v^2 + w^2), and the analogous z flux.
  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: The remaining volume term is a consistent and high-order accurate approximation of the continuous kinetic-energy balance with conservative advective terms.The argument is repeated for G#,1 and H#,1 before the final balance is stated.
  • Appendix A.2. Proof of Thm. 2 [Kinetic energy preservation]: The proof concludes that the discrete volume term reproduces the continuous kinetic-energy balance structure.The pressure contribution appears through the consistent high-order approximation involving the numerical pressure trace.

Appendix B. Curvilinear flux differencing form

The appendix extends flux-differencing DGSEM formulations to curved hexahedral elements by mapping the conservation law to reference coordinates and incorporating metric terms. It explains how split-form contravariant fluxes address aliasing and preserve high-order accuracy and conservation.

  • Appendix B. Curvilinear flux differencing form: Curved hexahedral elements are mapped from a reference cube to the physical domain using a polynomial transformation X(ξ).The mapped conservation law is expressed in reference coordinates.
  • Appendix B. Curvilinear flux differencing form: Covariant basis vectors generate contravariant basis vectors scaled by the Jacobian, which provide the metric terms for the mapped operators.The relation Jai = J∇ξi = aj × ak defines the cross-product construction.
  • Appendix B. Curvilinear flux differencing form: An explicitly divergence-free contravariant basis-vector formulation is available in addition to the cross-product construction.The divergence-free form is particularly important on curved-sided hexahedral elements, whereas the cross-product form suffices for straight-sided meshes and two-dimensional problems.
  • Appendix B. Curvilinear flux differencing form: The mapped DGSEM uses collocated nonlinear fluxes and strong-form differentiation at Gauss–Lobatto nodes.The componentwise mapped system includes contravariant flux contributions in the three coordinate directions.
  • Appendix B.1. Split form stabilisation for curvilinear DGSEM: Curvilinear metric terms increase the nonlinearity of contravariant fluxes and create an additional source of aliasing-driven nonlinear instabilities.The appendix therefore replaces the standard volume discretisation with a split-form flux-differencing form.
  • Appendix B.1. Split form stabilisation for curvilinear DGSEM: Fisher's curvilinear extension supplies high-order flux-difference operators using contravariant operators in each coordinate direction.The volume flux functions are consistent and symmetric.
  • Appendix B.1. Split form stabilisation for curvilinear DGSEM: The resulting curvilinear approximation is high-order accurate and conservative in the Lax–Wendroff sense.The split-form volume discretisation replaces the standard discretisation while retaining the discussed surface-flux choices.
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