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Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy

Henry M. Haase, Andrew R. Winters, Jan Glaubitz

arXiv:2610.30272v1math.NAcs.NA

TL;DR

Flux differencing has been developed for polynomial-based SBP operators, while systematic accuracy analysis for FSBP operators has been lacking. This paper studies flux differencing and design order for general function spaces, finding that suitable non-polynomial spaces can match polynomial design order of equal dimension.

  • Problem

    Flux differencing has been developed for polynomial-based SBP operators, while systematic analysis of FSBP accuracy and design order is lacking.

    Source: FSBP operators for general function… · p. 3

  • Method

    The paper characterizes FSBP design order through a Wronskian-based criterion and analyzes flux differencing using the discrete chain rule when the space contains constants.

    Source: Our contribution · p. 3, §3.3 · p. 8

  • Results

    Suitable non-polynomial spaces achieve the design order of equal-dimension polynomial spaces, and flux-differencing semi-discretization retains the FSBP design order for smooth solutions.

    Source: Our contribution · p. 3

  • Takeaways & Limitations

    Tailoring the function space can substantially reduce the error constant while retaining design order for smooth solutions.

    Source: Our contribution · p. 3

    The framework is restricted to closed operators; evolving the function space in time requires an entropy-stable projection between bases, left for future work.

    Source: §6 · p. 25

Abstract

from arXiv · show

Summation-by-parts (SBP) operators are a cornerstone of provably stable high-order methods for hyperbolic conservation laws. For nonlinear problems, the flux-differencing framework of Fisher and Carpenter combines such SBP operators with entropy-conservative two-point fluxes to obtain semi-discretizations that are conservative in the sense of Lax--Wendroff and discretely entropy-conservative or stable. So far, this framework has been developed exclusively for operators that are exact for polynomials. Function-space SBP (FSBP) operators lift this restriction by being exact on a general finite-dimensional function space that can be tailored to the problem at hand. In this work, we extend flux differencing to FSBP operators for general (potentially non-polynomial) function spaces. We prove that, whenever the operator has a diagonal norm and its function space contains the constants, the resulting semi-discretization is conservative in the sense of Lax--Wendroff, semi-discretely entropy-conservative within a single element, and entropy-stable across element interfaces. We also provide the first theoretical accuracy analysis of FSBP operators: We characterize their design order through an algebraic criterion on the underlying function space, prove that suitable non-polynomial spaces achieve the same design order as polynomial spaces of equal dimension, and show that the flux-differencing semi-discretization retains this order. Notably, the entropy-conservative two-point flux is reused unchanged; only the underlying function space differs from the polynomial case. Numerical experiments for the Burgers' equation and the compressible Euler equations in one and two dimensions confirm these properties.

1. Introduction.

The paper extends flux differencing from polynomial SBP operators to FSBP operators exact on general function spaces, addressing both entropy guarantees and the previously unanalyzed accuracy of FSBP operators. Its analysis establishes conditions for conservation and entropy stability, characterizes design order, and shows that flux differencing retains that order.

SBP operators.

  • SBP operators mimic integration by parts discretely, moving volume contributions in the energy balance to element boundaries for control through interface and boundary terms.This underpins their use in provably stable high-order semi-discretizations.Source: SBP operators · pp. 1–2

Entropy stability and flux differencing.

  • Flux differencing combines a diagonal-norm SBP operator with an entropy-conservative two-point flux to obtain a conservative, discretely entropy-conservative scheme, made entropy-stable by interface dissipation.The framework is systematic and operator independent, and has been extended beyond collocated Gauss–Lobatto nodes.Source: Entropy stability and flux differencing · p. 2

FSBP operators for general function spaces.

  • FSBP operators replace polynomial exactness with exactness on a prescribed finite-dimensional function space, which can be tailored to solution behavior such as boundary layers or oscillations.For example, exponentials or trigonometric functions may represent such solutions better than polynomials.Source: FSBP operators for general function… · p. 2, §2.3 · p. 5
  • The paper addresses two gaps: flux differencing had been developed only for polynomial-based operators, and FSBP design order lacked systematic theoretical analysis.FSBP operators exist on a grid when a positive quadrature rule is exact for the derivative product space.Source: FSBP operators for general function… · p. 3, §2.3 · p. 6

Our contribution.

  • With a diagonal norm and constants in the function space, FSBP flux differencing is Lax–Wendroff conservative; with an entropy-conservative two-point flux, it is entropy-conservative within an element and entropy-stable across interfaces.The two-point flux is reused unchanged; only the function space differs from the polynomial case.Source: Our contribution · p. 3
  • The accuracy analysis bounds truncation error by simultaneous approximation of a function and its derivative, characterizes design order through a Wronskian-based criterion, and shows flux differencing retains that order for smooth solutions.Suitable non-polynomial spaces attain the design order of equal-dimension polynomial spaces, while tailoring can reduce the error constant.Source: Our contribution · p. 3
  • Experiments for Burgers’ equation and one- and two-dimensional compressible Euler equations demonstrate discrete conservation, entropy conservation, entropy stability, and improved accuracy when non-polynomial spaces better represent solutions.The paper also formulates a spatially varying implementation that permits different spaces, nodes, and operators in each element.Source: Our contribution · p. 3

3. Flux differencing for FSBP operators.

Flux differencing extends to general FSBP spaces, with conservation and entropy properties under diagonal-norm and constant-inclusion assumptions. The same framework couples elements through entropy-conservative or entropy-stable interface fluxes.

3.1. The flux-differencing semi-discretization.

  • Flux differencing extends from polynomial SBP operators to FSBP operators for general, potentially non-polynomial spaces, using a consistent symmetric entropy-conservative volume flux.The semi-discretization replaces the derivative of nodal flux values with a two-point operation.Source: §3 · p. 6, §3.1 · p. 6

3.2. Conservation on a single element.

  • With a diagonal norm and constants in the function space, the weighted solution changes only through surface fluxes, giving Lax–Wendroff conservation.The diagonal norm enables entrywise contraction, while D1 = 0 makes the volume term telescope to boundary fluxes.Source: §3.2 · p. 7

3.3. Entropy conservation on a single element.

  • With a diagonal-norm FSBP operator containing constants and an entropy-conservative volume flux, discrete entropy changes only through surface fluxes.The volume term telescopes to an entropy-flux boundary difference; the surface flux determines whether total entropy is conserved or dissipated.Source: §3.3 · pp. 7–8

3.4. Entropy stability and element coupling.

  • Across elements, entropy-conservative interface fluxes leave total entropy changing only at physical boundaries, while entropy-stable fluxes bound it by physical boundary fluxes.The result applies when each element has a diagonal-norm FSBP operator containing constants and the elements share an interface flux.Source: §3.4 · p. 8

4. Accuracy of FSBP operators.

The accuracy analysis characterizes FSBP design order through the function space and shows that flux differencing retains that order. Suitable non-polynomial spaces match equal-dimensional polynomial spaces, while tailoring can reduce error constants.

4. Accuracy of FSBP operators.

  • For norm-bounded F-exact operators, design order p is equivalent to differentiating every polynomial through degree p with truncation error O(h^p).The function-space criterion uses the Taylor polynomials of a basis; the cited related criterion is Wronskian-based.Source: §4 · p. 10
  • Suitable non-polynomial spaces achieve design order p = dim F − 1, matching polynomial spaces of equal dimension.The exponential space E_d achieves p = d for every nonzero α, matching P_d.Source: §4 · pp. 10–11
  • The accuracy bounds concern truncation errors; they do not alone establish convergence rates for the solution error.The global discrete L2 estimate preserves order p under the stated elementwise bounds.Source: §4 · p. 11
  • A three-dimensional space with a centered sine term supports only design order p = 1, whereas replacing that term with cosine restores p = 2.In the sine example, degree-2 Taylor polynomials span only P_1.Source: §4 · p. 11
  • Tailoring the function space can reduce error constants without increasing design order, and enriching it with a steady solution can reduce error near that solution.The relevant approximation concerns nonlinear compositions in the flux-differencing volume term.Source: §4 · pp. 11–12
  • For smooth solutions, the flux-differencing volume term and semi-discretization inherit the underlying FSBP operator's design order.The volume-term result assumes a consistent, symmetric two-point flux that is continuously differentiable near the diagonal.Source: §4 · p. 12

5. Computational examples.

The numerical experiments investigate conservation, entropy properties, and volume-term design-order accuracy.

5. Computational examples.

  • The experiments investigate the scheme's discrete mass conservation properties.They are designed to test Corollary 3.3.Source: §5 · p. 12
  • The experiments investigate semi-discrete entropy conservation and stability.They target the properties established in Theorems 3.4 and 3.5.Source: §5 · p. 12
  • The experiments investigate the design-order accuracy of the volume term.This tests the property analyzed in Proposition 4.10.Source: §5 · p. 12

Implementation and diagnostics.

The computational experiments are reproducible through a public repository that extends Trixi.jl for spatially varying FSBP operators.

Implementation and diagnostics.

Function spaces and operators.

The operators use polynomial, exponential, and trigonometric approximation spaces, with constants included and quadrature and grids constructed for each space.

Function spaces and operators.

  • The polynomial baseline P_d recovers classical polynomial SBP operators, while the FSBP operators use exponential and trigonometric spaces.Each considered space contains the constant function.Source: Function spaces and operators · p. 13
  • Closed Gaussian FSBP operators use boundary-inclusive grids and positive, (FF)′-exact quadrature for the norm matrix.The generalized Gaussian quadrature is computed with modified algorithms.Source: Function spaces and operators · p. 13
  • Because the considered spaces are translation invariant, operators need only be computed for different element lengths, not separately for each element location.A non-translation-invariant space may require a different operator on each element.Source: Function spaces and operators · p. 13

Two-point fluxes.

Burgers’ and Euler equations use entropy-conservative volume fluxes, with dissipation added to form entropy-stable interface fluxes.

Two-point fluxes.

  • For Burgers’ equation, the volume flux is the entropy-conservative two-point flux (uL^2 + uLuR + uR^2)/6, with entropy variable w = u.The flux satisfies the entropy-conservation condition (2.5).Source: Two-point fluxes · p. 13
  • For compressible Euler, the entropy-conservative and kinetic-energy-preserving flux of is used in both volume and interface terms.Source: Two-point fluxes · p. 13
  • Entropy-stable interface fluxes add matrix dissipation on the entropy-variable jump, using modified Roe-type dissipation for Euler or local Lax–Friedrichs dissipation for Burgers.Source: Two-point fluxes · p. 13

Time integration.

The experiments compare time integrators and structure-preserving diagnostics, then test Burgers’ equation before and after shock formation. Before the shock, the FSBP spaces show higher observed convergence orders than their dimension-based guarantees, while mass and entropy behave as predicted.

Time integration.

  • Relaxation Runge–Kutta time integration is used for entropy-conservation tests, while ninth-order Verner and SSP-RK methods serve convergence and robustness studies.Time steps are chosen so temporal error does not contaminate reported spatial quantities.Source: Time integration · p. 13

Diagnostics.

  • The diagnostics track relative error in the discrete P-norm, experimental convergence order, discrete mass and total entropy, and residuals against predicted balances.For periodic source-free tests, mass should remain constant and entropy should be constant for entropy-conservative fluxes or non-increasing for entropy-stable fluxes.Source: Diagnostics · pp. 13–14

5.1. Burgers’ equation.

  • For inviscid Burgers’ equation with square entropy on periodic domain [−1, 1], a shock forms at t∗=1/π; E2, P2, and P3 solutions on 16 elements are compared with the exact solution before and after shock formation.Source: §5.1 · p. 14, Figure 1 · p. 15

Smooth regime.

  • E2 and E∗ reach EOCs of approximately 4 and 5, respectively, while polynomial P_d operators reach approximately d + 1 before shock formation.The FSBP observations exceed the dimension-based guarantees; the convergence study compares these spaces with polynomial operators.Source: Smooth regime · pp. 15–16, Figure 2 · p. 16
  • The observed FSBP orders exceed the guarantees of dim F − 1, leaving their theoretical explanation for future work.The experiments indicate order N − 1 for Gaussian FSBP operators, based on node count.Source: Smooth regime · pp. 15–16
  • On a periodic domain, discrete mass is conserved to high precision, with errors below 10^-13; P4 and E∗ show a small upward drift near 10^-14.P2, P3, and E2 remain at machine precision.Source: Smooth regime · p. 16
  • With relaxation time integration, total entropy is conserved up to round-off for entropy-conservative fluxes and is monotonically non-increasing for entropy-stable fluxes.The entropy-conservative scheme also continues past shock formation while preserving entropy, although this is physically unsound.Source: Time integration · p. 13, Smooth regime · p. 16

Shock regime.

In the Burgers’ manufactured-solution test, the tailored E2 operator is more accurate than equal-cost P3, while E* reaches machine precision for α = 1. The experiments also report conservation and entropy behavior for a compressible Euler density wave, convergence comparisons, and a modified Sod shock setup.

Shock regime.

  • At t = 1.5t∗, P3 and E2 are nearly indistinguishable and track the Burgers’ shock well; entropy-conservative solutions are highly oscillatory and do not match the exact solution.P2 behaves similarly to P3 and E2, with slightly different oscillations around the shock.Source: Shock regime · pp. 16–17

5.2. Manufactured solution for Burgers’ equation.

  • For α = 1, E* produces a Burgers’ solution at machine precision, while E2 is uniformly more accurate than equal-cost P3 by a factor of ≈2.29.E2 attains fourth-order convergence, matching cubic P3; E* is almost at machine precision from the start.Source: §5.2 · p. 18, Table 2 · p. 19

Conservation and entropy.

  • For the smooth periodic Euler density wave, discrete mass errors stay small, though P4 and E* reach 10^-13 by t = 100; higher-precision runs reduce the errors.The authors suspect floating-point inaccuracies and accumulated rounding errors cause the increase.Source: Conservation and entropy · p. 20
  • For the Euler density wave, total entropy errors remain bounded at approximately 10^-12, consistent with Theorem 3.5.Source: Conservation and entropy · p. 20

Accuracy.

  • For Euler density-wave convergence, degree-d polynomial SBP and same-node-count FSBP operators have asymptotic EOC d + 1; T* has smaller error at the same EOC.P3 and E2 with four nodes, and P4 and E* with five nodes, show almost identical convergence behavior.Source: Accuracy · p. 20
  • For the Euler density wave, logarithmic means in the entropy-conservative flux move the numerical-flux derivative outside T*, explaining why exact solution representation does not yield the Burgers’ E* result.Using standard weak-form DG instead of entropy-conservative flux differencing retains a semi-discretization accurate to machine precision.Source: Accuracy · pp. 20–21

5.5. Modified Sod shock tube.

  • The modified Sod test uses a transonic left rarefaction with a sonic point; entropy-condition violations can produce an entropy glitch or expansion shock there.No wave reaches a boundary before t = 1/5, and this is the only test with nonzero σS in (5.9).Source: §5.5 · p. 22

Robustness.

The FSBP flux-differencing schemes capture the Sod shock and contact discontinuity and remain stable in a demanding two-dimensional test, while entropy dissipation at shocks is not reproduced exactly. The outlook identifies open accuracy questions and extensions beyond the closed-operator setting.

Robustness.

  • The Sod shock-tube solutions capture the shock and contact discontinuity at the correct location, but both show oscillations.Figure 10 compares density, momentum, and total energy against the exact Riemann solution for P3 and E2.Source: Robustness · p. 22, Figure 10 · p. 23

Conservation.

  • The E2 conservation error is comparable to P3, but its entropy residual exceeds 10^-4 because entropy is dissipated only at element boundaries.The schemes do not match the exact solution’s entropy dissipation at shocks, allowing entropy over- or underproduction.Source: Conservation · p. 22, Figure 11 · p. 24

5.6. Two-dimensional compressible Euler equations.

  • The two-dimensional Euler extension applies one-dimensional FSBP operators dimension by dimension on a tensor-product Cartesian grid.Source: §5.6 · p. 22

Kelvin–Helmholtz instability.

  • Both operators remain stable through t = 3 in the Kelvin–Helmholtz test, but fail just before t ≈3.6; weak-form runs crash earlier.On this mesh, the weak-form P3 and E2 runs crash at approximately t = 2.25 and t = 2.05, respectively.Source: Kelvin–Helmholtz instability · p. 23, Figure 12 · p. 25

6. Summary and outlook.

  • With a diagonal norm and constants in F, the method is conservative, entropy-conservative within one element, and entropy-stable across interfaces; its accuracy analysis characterizes design order and shows flux differencing retains it.Suitable non-polynomial spaces can match the design order of equal-dimensional polynomial spaces, and tailoring F can reduce the error constant.Source: §6 · pp. 23–24
  • A full convergence analysis remains open, and Gaussian FSBP operators converge one order faster than the design order guaranteed by the Wronskian criterion.The guarantee is dim F −1, while the observed rate is N −1.Source: §6 · pp. 24–25
  • The study uses closed operators; evolving the function space in time requires an entropy-stable projection between bases, while curved meshes also remain an open extension.The paper also notes that curved-mesh metric terms and transformed spaces pose additional design choices for FSBP operators.Source: §6 · p. 25

Appendix A. Notation.

The notation distinguishes nodal values from vector-valued states and specifies how indices and operators act on the discrete solution.

Appendix A. Notation.

  • Bold upright symbols denote nodal vectors, while bold italic symbols denote vector-valued functions or states; subscripts index nodes and, in multi-element settings, elements.For an m-component system, the discrete solution is an N × m array whose row at node i is the state there.Source: Appendix A · p. 26

Appendix B. Proof of Corollary 3.3.

The proof establishes a telescoping, subcell-flux form for the FSBP volume term and uses it to derive conservation. Constants in the function space ensure consistent subcell fluxes and zero row sums for Q.

Appendix B. Proof of Corollary 3.3.

  • The proof rewrites the volume term using symmetric two-point fluxes and the off-diagonal skew identity for Q, yielding a telescoping subcell-flux representation.Source: Appendix B · p. 27
  • When the nodal state is constant, consistency of the two-point flux and 1 ∈ F make each subcell flux equal the physical flux.The condition 1 ∈ F gives Q1 = 0, so each row of Q sums to zero.Source: Appendix B · pp. 27–28
  • Summing the semi-discrete scheme with the diagonal norm and applying the SBP identity reduces the volume contribution to boundary fluxes, yielding the conservation assertion.Source: Appendix B · p. 28

Appendix C. Proof of Theorem 3.4.

The entropy proof contracts the volume term with entropy variables and uses the entropy-conservative flux identity to turn the volume contribution into a boundary entropy-flux balance.

Appendix C. Proof of Theorem 3.4.

  • Contracting the semi-discrete scheme with the entropy variables and summing over nodes groups the boundary contributions into the entropy potential, yielding the single-element entropy assertion.Source: Appendix C · pp. 28–29
  • The volume entropy identity applies to a diagonal-norm FSBP operator containing constants and an entropy-conservative two-point flux, converting the contracted volume sum into boundary terms.Source: Appendix C · pp. 28–29
  • The resulting identity is the discrete counterpart of integration by parts for the entropy-variable-weighted flux.Source: Appendix C · p. 29
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