Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and...

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Fluid Dynamics

Background:

• The discontinuous Galerkin (DG) method is a powerful numerical technique for solving differential equations.
• Hyperbolic conservation laws model phenomena like fluid flow and wave propagation.
• Accuracy enhancement is crucial for reliable simulations of nonlinear systems.

Purpose of the Study:

• To analyze accuracy enhancement for the DG method applied to 1D nonlinear hyperbolic conservation laws.
• To investigate the superconvergence properties of DG solutions.
• To extend existing accuracy-improving techniques to nonlinear problems.

Main Methods:

• Analysis of divided differences of DG errors in various norms.
• Application of duality arguments to derive superconvergence results.
• Extension of the Smoothness-Increasing Accuracy-Conserving (SIAC) filter.
• Numerical experiments to validate theoretical findings.

Main Results:

• Proved that the k-th order divided difference of the DG error is of order O(h^p) in the L^2 norm using upwind fluxes.
• Derived superconvergence results of order O(h^(p+1)) in the negative-order norm via duality.
• Demonstrated that the SIAC filter can achieve at least (p+1)-th order superconvergence for post-processed solutions.
• Provided explicit proofs for optimal convergence rates for variable coefficient hyperbolic equations.

Conclusions:

• The DG method can achieve enhanced accuracy for nonlinear hyperbolic conservation laws.
• Superconvergence results are attainable through specific error analysis and post-processing techniques.
• The findings are validated by numerical experiments, confirming the theoretical advancements.