The error estimate of entropy-stable discontinuous Galerkin methods for hyperbolic conservation laws

Technical Summary: Error Estimates of Entropy-Stable Discontinuous Galerkin Methods for Hyperbolic Conservation Laws

Problem Statement
Hyperbolic conservation laws govern physical phenomena such as gas dynamics, where solutions often develop shocks and contact discontinuities in finite time. To ensure the well-posedness of these problems, entropy inequalities are required to select the physically admissible weak solution. While Chen and Shu [J. Comput. Phys. 345 (2017)] established a unified framework for constructing high-order entropy-stable discontinuous Galerkin (ESDG) methods using specific numerical quadrature and summation-by-parts (SBP) operators, a rigorous convergence theory was missing. Existing error analyses for standard DG methods rely on the assumption that the numerical solution resides in a prescribed space of piecewise polynomials. However, for the nodal ESDG method, the nodal values produced by the scheme do not generally correspond to a piecewise polynomial of degree $k$ in multiple space dimensions. Consequently, classical error estimates are not directly applicable, leaving a gap in the theoretical justification for the convergence of these entropy-stable schemes.

Methodology
This paper establishes rigorous a priori error estimates for semi-discrete ESDG methods on general unstructured meshes. The analysis employs a finite-difference-type consistency–stability argument formulated directly at the nodal level, rather than relying on standard variational arguments.

The core methodology involves:

1. Polynomial Reconstruction Hypothesis: The authors assume that for each element, the nodal vector admits a unique polynomial reconstruction in a finite-dimensional space $V_h(K)$ containing polynomials of degree $k$. This allows the use of standard inverse estimates and approximation theory.
2. $L^\infty$ A Priori Bound: The analysis assumes a bound on the error in the $L^\infty$ norm, which is justified via a standard continuity argument (continuation method).
3. Consistency-Stability Framework:
   • Consistency: A local truncation error is derived by inserting the exact solution's nodal values into the semi-discrete scheme.
   • Stability: An error equation is formulated by subtracting the numerical scheme from the exact solution's evolution. The error is measured in a quadrature-based norm equivalent to the broken $L^2$ norm on the reconstruction space.
   • Energy Estimates: The error equation is multiplied by the entropy variable error and integrated. The volume terms are controlled using the skew-symmetric SBP structure combined with entropy symmetrization of the flux Jacobians. Interface contributions are managed via the entropy stability of the numerical flux. Nonlinear terms are estimated using Taylor expansions and mesh-scaled matrix-norm bounds.
   • Closure: The resulting differential inequality is closed using a Gronwall argument.

The framework is extended to the Entropy-Stable Oscillation-Free DG (ESOFDG) method, which includes additional damping terms to suppress spurious oscillations. The authors prove that these damping terms do not degrade the convergence order.

Key Contributions

• Rigorous Convergence Theory: The paper provides the first rigorous a priori error estimate for the semi-discrete ESDG method introduced by Chen and Shu [6], proving an $O(h^k)$ error estimate in a quadrature-based norm for smooth solutions of both scalar conservation laws and systems.
• Resolution of the Polynomial Assumption Gap: By introducing the polynomial reconstruction hypothesis, the authors bridge the gap between the nodal flux-differencing formulation (which violates the standard piecewise polynomial assumption) and classical error analysis techniques.
• Extension to ESOFDG: The analysis is successfully extended to the ESOFDG method, demonstrating that the additional damping terms, designed to control oscillations, preserve the theoretical convergence rate.
• Technical Lemmas: The paper establishes necessary technical results regarding the scaling properties of SBP operators, local truncation errors, and bounds on the damping coefficients for the ESOFDG scheme.

Results

• Theoretical Bounds: Under the assumptions of polynomial reconstruction and an $L^\infty$ a priori bound, the paper proves that the error in the quadrature-based norm scales as $O(h^k)$ for polynomial degree $k$. This result holds for both scalar equations and systems of conservation laws.
• Numerical Experiments: Numerical tests on the Burgers equation (1D and 2D) and the 2D Euler equations (vortex evolution) confirm the theoretical findings.
  • The observed convergence rates generally align with or slightly exceed the theoretical $O(h^k)$ prediction.
  • In some cases (e.g., 1D Burgers with $k=3$), the observed rates approach $O(h^{k+1})$, suggesting the theoretical bound may not be sharp.
  • The ESOFDG scheme exhibits convergence rates indistinguishable from the standard ESDG scheme, confirming that the damping terms do not sacrifice accuracy.

Significance and Claims
The authors claim that this work closes a critical gap in the convergence theory for entropy-stable DG schemes. By establishing a rigorous $O(h^k)$ error estimate, the paper provides a theoretical foundation for the reliability of these high-order methods in capturing physically admissible solutions.

The paper modestly acknowledges that while the numerical experiments suggest observed rates may exceed the theoretical bound by up to half an order (potentially indicating superconvergence or delicate cancellation mechanisms), the current analysis provides a solid, albeit potentially non-sharp, lower bound. The authors note that whether a sharper $(k + 1/2)$-th order estimate can be derived within the finite element variational framework remains an open question. The primary contribution is the establishment of a rigorous convergence theory for a class of schemes that were previously lacking such justification.