Summation-by-parts operators for correction procedure via reconstruction

Hendrik Ranocha; Philipp Öffner; Thomas Sonar

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Summation-by-parts operators for correction procedure via reconstruction

Hendrik Ranocha, Philipp Öffner, Thomas Sonar

arXiv:1511.02052v2Nov 6, 2015math.NA

TL;DR

CPR methods lacked published nonlinear entropy-stability proofs despite established linear-stability results. The paper reformulates CPR with SBP operators and SATs, proving conservation and stability in method-adapted discrete norms and entropy stability for Burgers’ equation with general SBP CPR methods.

Problem
Published CPR results established linear stability, but nonlinear entropy-stability proofs were not available to the authors’ knowledge.
Method
The paper reformulates general CPR methods using SBP operators and extends skew-symmetric conservation-law formulations with an additional correction term.
Results
The reformulation proves conservation and stability in discrete norms, while the extended formulation proves Burgers’ equation conservative and stable for general SBP CPR methods, including bases without boundary nodes.
Takeaways & Limitations
SBP CPR methods provide a unified framework recovering linearly stable schemes and extending provable entropy stability beyond the Lobatto-Legendre case.
Takeaways & Limitations
Explicit Euler and standard SSP time discretisations do not yield the desired fully discrete stability estimate in the reported straightforward analysis.

Abstract

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The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.

1. Introduction

The paper places CPR methods within the broader effort to make high-order conservation-law schemes more accurate without sacrificing robustness. It reformulates CPR with SBP operators to establish conservation and stability, including nonlinear results for Burgers’ equation.

High-order methods can provide more accurate solutions at the same computing cost, but are generally less robust and more complicated than low-order methods.
Nonlinear stability analysis for CPR methods is less developed than linear stability analysis, despite existing linearly stable schemes.
SBP operators with SATs provide a framework for weak boundary enforcement and discrete analogues of continuous stability arguments.
The paper applies the SBP framework to general CPR methods and demonstrates conservation and stability, extending the framework to a nonlinear case.
For Burgers’ equation, the paper proves discrete conservation and stability for a skew-symmetric formulation with Lobatto-Legendre nodes.
An additional correction-term formulation yields provable conservation and stability for general SBP CPR methods, with numerical tests confirming the theoretical results.

2. Existing formulations for SBP operators and CPR methods

The section introduces SBP operators and CPR methods as semidiscrete frameworks for hyperbolic conservation laws. CPR computes flux derivatives with boundary corrections, while SBP structure and SATs provide discrete integration-by-parts and weak boundary enforcement.

Existing formulations for SBP operators and CPR methods: SBP finite-difference methods and CPR schemes are designed as semidiscretisations of hyperbolic conservation laws with initial and boundary conditions.
2.1. SBP schemes: Standard SBP operators use nodal solution values, a differentiation matrix, and a positive quadrature-based norm to approximate derivatives and continuous energy measures.
2.1. SBP schemes: SATs impose boundary and inter-block conditions weakly through differences between desired and given boundary values.
2.2. CPR methods: The one-dimensional FR approach maps elements to [-1,1], evaluates fluxes at nodes, and constructs a semidiscrete derivative using polynomial bases.
2.2. CPR methods: CPR corrects the interpolated flux with degree-p+1 boundary functions weighted by differences between physical and numerical boundary fluxes.
2.2. CPR methods: The LCP approach extends this construction to triangles through correction terms formed from boundary differences between common numerical fluxes and local flux values.
2.2. CPR methods: FR and LCP are treated under the common CPR name because the approaches are similar and can be reformulated into one another.

3. CPR methods using SBP operators: Linear advection

The paper formulates CPR semidiscretisations with differentiation, restriction, and correction matrices, then connects suitable correction choices to SBP and DGSEM structures. Constant-velocity linear advection provides the setting for examining conservation and linear stability.

The section introduces a CPR formulation for constant-velocity linear advection to investigate linear stability and conservation.
In matrix form, the semidiscretisation combines the derivative Df with a correction term C(f_num−Rf), where R restricts values to element boundaries.
A CPR method is parameterized by a local basis, its derivative and restriction operators, and a correction matrix adapted to that basis.
SBP CPR representations associate nodal bases with positive quadrature weights, a volume norm M, and a boundary quadrature satisfying the SBP property.
The SBP property mimics integration by parts discretely through the relation MD+D^TM=R^TBR.
For Gauß-Lobatto-Legendre nodes, the restriction and boundary matrices simplify, and the choice C=M^-1R^T B recovers the strong-form DGSEM formulation.

3.2. Conservation

The conservation analysis derives a discrete balance law by multiplying the semidiscrete scheme by the constant function and applying the SBP identity. Conservation follows when the correction operator satisfies a specific compatibility condition, including under periodic assembly.

For polynomial bases and quadrature exact through degree 2p−1, the SBP property holds automatically for the relevant polynomial products.
The discrete conservation derivation multiplies the semidiscrete equation by the constant function and applies the SBP property and exactness of constant differentiation.
The resulting balance contains volume, boundary, and correction contributions that can be expressed using the restriction matrix and numerical flux.
The CPR scheme is conservative across elements when the correction operator satisfies 1^TMC=1^TR^TB.
With periodic boundary conditions, matching numerical fluxes at shared interfaces produce global conservation after summing element contributions.
The conservation result extends prior conservation proofs for diagonal-norm SBP operators that include boundary nodes.

3.3. Linear stability

The SBP formulation derives linear stability by estimating a discrete norm through boundary terms and selecting correction and matrix conditions that eliminate the remaining interior contribution.

3.3. Linear stability: For linear advection, α ≥0 ensures d dt ∥u∥M ≤0, giving stability through a discrete energy estimate.The parameter choice includes central flux at α = 0 and fully upwind flux at α = 1.
3.3. Linear stability: The correction C = (M + K )−1R T B converts the norm-rate expression's second term into a boundary term.This leaves the term involving K D as the remaining quantity to control.
3.3. Linear stability: Choosing K D antisymmetric makes the remaining interior term vanish, so the same boundary estimates yield linear stability.
3.3. Linear stability: The resulting CPR method is linearly stable in the discrete norm ∥·∥M+K when M + K is positive definite and K D is antisymmetric.

3.4. Symmetry

The symmetry analysis identifies conditions under which SBP CPR correction functions are symmetric, linking matrix commutation properties to the reflection symmetry of the correction procedure.

3.4. Symmetry: Identifying the correction matrix with derivatives of gL and gR yields the reflection relation gR(ξ) = gL(−ξ), avoiding directional bias.
3.4. Symmetry: The SBP CPR method is symmetric when the modal matrix condition ˆJ ( ˆ M + ˆK ) = ( ˆ M + ˆK ) ˆJ holds.
3.4. Symmetry: For quadrature exact through degree 2p−1, positive definite M + K and K D +D T K T = 0 imply the required commutation condition.
3.4. Symmetry: The proof establishes the commutation relation by induction on the polynomial degree using the alternating structure of the Legendre basis.

3.5. Summary

The theorem summarizes conservation and linear stability for one-dimensional SBP CPR methods under quadrature, mass, restriction, boundary, and derivative-matrix conditions.

3.5. Summary: The theorem assumes a positive diagonal mass matrix, boundary restriction operator, boundary matrix B = diag(−1, 1), and derivative matrix satisfying M D + D T M = R T B R.
3.5. Summary: If 1T M C = 1T R T B, the SBP CPR method is conservative.
3.5. Summary: If C = (M + K )−1R T B, M + K is positive definite, and K D is antisymmetric, the method is linearly stable in ∥·∥M+K.

3.6. The one parameter family of Vincent et al. (2011)

The one-parameter Vincent family is recovered by choosing a correction matrix based on K = κ(D p)T M D p and translating between SBP CPR and FR parameters.

3.6. The one parameter family of Vincent et al. (2011): The ansatz K = κ(D p)T M D p enforces the discrete-norm formulation and transforms consistently under basis changes.
3.6. The one parameter family of Vincent et al. (2011): The associated SBP CPR methods are conservative and linearly stable when κ satisfies the positivity conditions in equation (49).
3.6. The one parameter family of Vincent et al. (2011): The parameter relation c = 2κG = 2κL + cHu connects the SBP parameter to Vincent et al.'s parameter and Huynh's g2 scheme.
3.6. The one parameter family of Vincent et al. (2011): Huynh's g2 scheme coincides with the DGSEM scheme using Lobatto-Legendre nodes and a lumped mass matrix.

3.7. The multi parameter family of Vincent et al. (2015)

The multiparameter Vincent et al. family extends the one-parameter linearly stable and conservative schemes, while incorporating solution-point coordinates into the SBP CPR analysis.

The multiparameter family produces results similar to the one-parameter family because the latter is contained within the extended scheme range.
Unlike the earlier analysis, solution-point coordinates are treated as important SBP CPR parameters, motivating investigation of discrete norms and stability.
Using Lobatto-Legendre quadrature transforms the parameter space but recovers the same schemes obtained with Gauß-Legendre points.

3.8. Numerical examples

Numerical experiments validate SBP CPR implementations for periodic linear advection and examine correction parameters, fluxes, quadrature norms, and convergence. The natural norm associated with each correction preserves or dissipates energy as predicted, while other quadratures can oscillate.

The experiments repeat prior linear-advection tests with 10 elements, order p = 3, periodic boundaries, and fourth-order Runge–Kutta time integration over ten traversals.
For central fluxes, the natural quadrature associated with each correction exactly conserves energy, whereas other quadratures produce bounded oscillations.The natural choices are Gauß for c = c0 and Lobatto for c = cHu.
With upwind fluxes, energy decays in the associated norm, and stronger dissipation appears for c = c−/2 and c = cHu.Other quadrature choices still show oscillations while decaying overall.
Convergence studies compare Gauß- and Lobatto-Legendre bases across polynomial degree and element count for multiple correction parameters.The studies use upwind fluxes and report L2 errors at u(20).
For the natural choice κ = 0, the Gauß-Legendre basis is clearly superior when the polynomial degree is fixed and the number of elements varies.

3.9. Influence of time discretisation

The time-discretisation analysis examines explicit Euler applied to an SBP CPR semidiscretisation and finds that the time-step quadratic term obstructs a straightforward fully discrete stability estimate.

Explicit Euler expands the discrete norm into the original norm, a semidiscrete contribution, and a nonnegative time-step-squared term.
The semidiscrete estimate controls the first-order contribution, but the positive quadratic term can cause discrete-norm growth and stability issues.
The resulting estimate contains volume terms that prevent a straightforward fully discrete stability proof for explicit Euler.The same nonpositive stability conclusion extends directly to standard SSP methods formed from convex combinations of Euler steps.

4. CPR methods using SBP operators: Burgers’ equation

For Burgers’ equation, the paper extends skew-symmetric CPR formulations with correction terms to prove conservation and stability for Lobatto-Legendre and general SBP bases, with numerical tests confirming the theory.

With Lobatto-Legendre nodes, the skew-symmetric CPR method is stable in the discrete M-norm, but this relies on boundary nodes representing squared restrictions exactly.This property does not generally hold for Gauss-Legendre nodes.
For general SBP bases without boundary nodes, additional corrections to the divergence and boundary terms are required to obtain stability.The authors identify this correction as a new extension of the skew-symmetric formulation.
The corrected SBP CPR formulation is both conservative and stable in the discrete norm for numerical fluxes satisfying the stated condition.The admissible fluxes include ECON, local Lax-Friedrichs, and Osher’s flux.
The ECON flux conserves discrete momentum and energy to order 10^-5 but produces highly oscillatory, physically irrelevant solutions after discontinuities form.Energy conservation alone is unsuitable after shocks because energy is also an entropy for Burgers’ equation.
Local Lax-Friedrichs and Osher fluxes preserve momentum, dissipate energy after shock formation, and remain stable despite bounded shock-region oscillations.The corresponding numerical solutions are reported as physically acceptable.
Omitting the boundary-restriction correction for a Gauss-Legendre basis causes loss of guaranteed conservation and stability, including energy blowup or momentum loss.The additional correction is therefore necessary in the tested formulation.

5. Discussion and summary

The paper unifies CPR and SBP-SAT frameworks, recovering conservation and stability results while extending entropy-stable formulations to more general SBP CPR methods. Numerical studies support the theoretical results, but extensions to different correction matrices, fully discrete schemes, and multidimensional settings remain open.

The SBP-SAT reformulation embeds linearly stable CPR schemes and proves conservation and stability in method-adapted discrete norms.The framework includes the linearly stable schemes of Vincent et al. and establishes their stability in discrete norms associated with the method.
An additional correction term extends skew-symmetric formulations to prove conservation and entropy stability for more general SBP CPR methods.The extension includes general SBP bases and covers entropy stability for nonlinear Burgers’ equation.
Extending stability proofs to different norms was unsuccessful, while more complex systems may be difficult without boundary nodes.The discussion recommends the canonical correction matrix and notes potential advantages of Gauß-Legendre or Lobatto-Legendre nodes in different settings.
The authors identify different correction matrices, fully discrete schemes, and inherently multidimensional SBP CPR bases as directions for future work.Future investigations include entropy-stable correction functions, time-discretization effects, and bases for tensor-product elements and simplices.
Numerical tests compare SBP CPR configurations using Gauß-Legendre and Lobatto-Legendre bases, correction terms, polynomial orders, and numerical fluxes.Figures 9–14 examine Burgers’ equation with 20 elements across multiple bases, orders, correction choices, and fluxes.