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A collocation scheme that is equivalent to discontinuous Galerkin discretizations

This paper demonstrates that a specific spectral collocation scheme with the summation-by-parts property produces semi-discretizations equivalent to discontinuous Galerkin methods using the same quadrature, while offering favorable spectral properties and providing strategies to handle nullspace modes and restore equivalence in entropy-stable formulations.

Original authors: Jason Hicken

Published 2026-05-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jason Hicken

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake a perfect cake (solving a complex math problem) in a very specific kitchen. You have two different recipes for doing this:

The "Discontinuous Galerkin" (DG) Recipe: This is a very famous, highly respected method. It's like a master chef who carefully measures ingredients, mixes them in a bowl, and then checks the result against a strict set of rules to ensure the cake rises perfectly. It's known for being stable and reliable.
The "Modal Collocation" (MC) Recipe: This is a newer, simpler-looking recipe introduced by a researcher named Chan. It looks different on paper—it seems to skip some of the careful mixing steps and just places ingredients directly into the pan at specific points.

The Big Discovery
The author of this paper, Jason Hicken, discovered something surprising: These two recipes produce the exact same cake.

Even though the MC recipe looks simpler and uses a different set of instructions, if you use the same measuring cups (quadrature points) as the DG recipe, the final result is mathematically identical. It's as if you took the master chef's perfect cake, looked at it from a different angle, and realized it was actually made using the simpler recipe all along.

The "Ghost" Ingredients (Nullspace)
There is a catch. The MC recipe sometimes adds extra "ghost" ingredients to the mix. In math terms, these are called "extraneous degrees of freedom."

The Good News: In a dynamic situation (like a cake baking and rising over time), these ghost ingredients are initialized to zero and stay zero. They are like invisible ghosts that don't affect the taste or texture of the cake. The math proves they just sit there doing nothing.
The Bad News: If you are trying to bake a cake that needs to sit perfectly still (a "steady" problem), these ghosts can cause the oven to malfunction (the math becomes singular or breaks).
The Fix: The paper suggests a simple "stabilizer" (Local-Projection Stabilization) that acts like a filter, removing these ghost ingredients if you need the cake to sit still.

The "Negative Weight" Surprise
Usually, when you measure ingredients, you use positive amounts (1 cup of flour, 2 eggs). However, the MC recipe can sometimes use "negative weights" (like -1/30 of a cup). In the world of standard math rules, this usually means the recipe is unstable and the cake will burn.
But because the MC recipe is secretly the same as the DG recipe (which is known to be stable), the paper shows that even with these weird negative measurements, the cake still turns out perfect. The MC method inherits the stability of its "twin," the DG method.

The "Entropy" Twist
The paper also tested a special version of the MC recipe designed to be "entropy conservative" (a fancy way of saying it preserves energy perfectly in a specific mathematical way).

The Problem: This special version didn't quite match the DG recipe perfectly. It was like a recipe that added a secret spice that the DG recipe didn't have. The results were slightly different.
The Solution: The author showed that if you take that special MC recipe and "project" (or filter) the results back through the standard mathematical lens, it snaps right back into being identical to the DG recipe.

Why Does This Matter?
The author asks: "If they are the same, which one should we use?"

For simple, linear problems, the DG method might still be faster because it's a well-oiled machine.
However, for complex, non-linear problems (like simulating shockwaves or turbulent air), the MC method might be easier to set up and run, especially on messy, triangular-shaped grids (like a pizza cut into uneven slices).

The Bottom Line
This paper is a "Rosetta Stone" for two different mathematical languages. It proves that a newer, simpler-looking method (Modal Collocation) is actually just a different way of writing the same thing as the gold-standard method (Discontinuous Galerkin). This gives scientists the confidence to use the simpler method, knowing it will produce the same high-quality, stable results, provided they handle a few specific edge cases (like the "ghost" ingredients in steady problems).

Technical Summary: A Collocation Scheme Equivalent to Discontinuous Galerkin Discretizations

Problem Statement
The paper addresses the construction of stable, high-order discretizations for partial differential equations (PDEs) on unstructured grids. While Summation-by-Parts (SBP) operators provide a robust framework for stability, multidimensional SBP operators on simplex elements (such as triangles) can be complex to construct and often lack closed-form expressions. Previous work by Chan introduced a projection-based SBP operator (referred to here as the Modal Collocation or MC operator) as an intermediate step in finite-element discretizations. However, the specific properties of using this operator directly in a collocation scheme, particularly its relationship to Discontinuous Galerkin (DG) methods, had not been fully explored. The central question is whether a semi-discretization based on this specific MC operator yields solutions equivalent to a modal DG discretization using the same underlying quadrature.

Methodology
The author defines the MC operator using orthonormal polynomial bases and positive-interior quadrature rules. The core methodology involves:

Operator Definition: Defining the first-derivative MC operator $D_{xd} = V_{xd}V^T W$ , where $V$ is the Vandermonde matrix of orthonormal basis functions, $V_{xd}$ contains their derivatives, and $W$ is the diagonal matrix of quadrature weights.
Theoretical Equivalence Proof: The paper constructs a proof demonstrating that the semi-discretization of a scalar nonlinear hyperbolic initial-boundary-value problem (IBVP) using the MC operator produces a solution vector $u(t)$ that is exactly the projection of the modal DG solution vector $\tilde{u}(t)$ onto the nodal space ( $u(t) = V\tilde{u}(t)$ ). This proof relies on decomposing the solution space into the polynomial subspace and the nullspace of the operator.
Spectral Analysis: The author analyzes the eigenvalues of the resulting system matrices. It is shown that the spectrum of the MC operator consists of the eigenvalues of the corresponding DG operator plus a set of zero eigenvalues associated with the nullspace.
Handling Nullspace Inconsistency: The paper acknowledges that MC operators are generally nullspace inconsistent (their nullspace contains more than just constant vectors). For unsteady problems, these extraneous modes are initialized to zero and remain zero. For steady problems, the author proposes Local-Projection Stabilization (LPS) to remove the nullspace singularity without affecting polynomial exactness.
Numerical Verification: The theory is verified through numerical experiments on triangular meshes using:
- The constant-coefficient advection equation.
- The nonlinear Burgers' equation.
- Comparisons between "Standard" MC discretizations and "Entropy-Conservative" (skew-symmetric) formulations.
- Tests involving quadrature rules with negative weights.

Key Contributions

Equivalence Theorem: The primary contribution is the proof that a modal collocation discretization using the MC operator is mathematically equivalent to a modal DG discretization, provided they share the same quadrature rule. This holds regardless of the number of degrees of freedom (quadrature points) in the collocation scheme.
Spectral Properties: The paper establishes that the spectral radius of the MC operator is effectively determined by the underlying DG method, with the addition of zero eigenvalues. This implies that the clustering of nodes (often problematic for spectral element methods on triangles) does not negatively impact the spectral radius of MC operators in the same way it does for traditional nodal SBP operators.
Nullspace Management: The work clarifies that while MC operators are nullspace inconsistent, this is not detrimental for time-dependent simulations if the initial conditions are projected correctly. For steady-state problems, the paper demonstrates how to recover consistency via projection-based numerical dissipation (LPS).
Entropy Conservation Nuance: The study reveals that entropy-conservative MC discretizations (based on skew-symmetric splittings) are not inherently equivalent to DG because the residual may not lie within the polynomial space. However, equivalence can be recovered by projecting the residual onto the polynomial basis.

Results

Constant-Coefficient Advection: Numerical results show that the $L_2$ difference between MC and modal DG solutions is at machine precision ( $\approx 10^{-15}$ ) for various polynomial degrees and quadrature exactness levels.
Spectral Radii: Eigenvalue distributions confirm that MC operators exhibit favorable spectral radii compared to min-norm SBP operators, which suffer from catastrophic growth due to node clustering on triangles.
Negative Weights: The equivalence holds even when the quadrature rule includes negative weights, a scenario where conventional SBP energy-stability theory might fail to predict stability. The stability is instead guaranteed by the equivalence to the energy-stable DG solution.
Burgers' Equation: For the nonlinear Burgers' equation, the "Standard" MC discretization is equivalent to DG. The "Entropy-Conservative" variant initially shows a discrepancy because the residual is not spanned by the polynomial basis. However, once the residual is projected onto the basis, equivalence is restored to machine precision.

Significance and Claims
The paper claims that the Modal Collocation operator introduced by Chan is not merely an intermediate tool but a valid discretization scheme that is effectively a modal DG method. The significance lies in the simplification of constructing high-order, stable schemes on unstructured simplex grids. By using the MC operator, one can leverage the closed-form expression of the operator while retaining the stability and spectral properties of DG methods.

The author modestly concludes that while the two methods are equivalent, the choice between them depends on the application. For linear problems where mass and stiffness matrices are precomputed, DG may be preferable. However, for nonlinear problems solved with explicit time-marching or matrix-free implicit schemes, the MC operator may offer computational advantages. The paper strongly advocates for the use of modal collocation as the preferred method for constructing multidimensional SBP operators.

Technical Summary: A Collocation Scheme Equivalent to Discontinuous Galerkin Discretizations

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