Superconvergence of the local discontinuous Galerkin method for nonlinear convection-diffusion problems

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• Nonlinear convection-diffusion equations are crucial in modeling various physical phenomena.
• Discontinuous Galerkin (DG) methods offer flexibility in handling complex geometries and solution properties.
• Superconvergence in numerical methods significantly enhances solution accuracy and efficiency.

Purpose of the Study:

• To analyze the superconvergence properties of local discontinuous Galerkin (LDG) methods for nonlinear convection-diffusion equations.
• To establish theoretical convergence rates for LDG methods under specific flux choices.
• To investigate the applicability of these findings to general flux functions.

Main Methods:

• Development and application of local discontinuous Galerkin (LDG) methods.
• Rigorous mathematical proof of superconvergence orders for specific flux functions (upwind and alternating).
• Validation through numerical experiments on arbitrary non-uniform regular meshes using piecewise polynomials of degree k.

Main Results:

• Proved [Formula: see text]th-order superconvergence of the numerical solution to a specific projection of the exact solution for LDG methods.
• Demonstrated the validity of the proof for piecewise polynomials of any degree k on non-uniform meshes.
• Numerical experiments confirmed that superconvergence holds for general flux functions, not just the specific ones used in the proof.

Conclusions:

• The local discontinuous Galerkin method exhibits significant superconvergence properties for nonlinear convection-diffusion problems.
• The theoretical findings are robust, applying to various polynomial degrees and mesh types.
• The observed superconvergence extends beyond the theoretically proven cases, suggesting broader applicability of LDG methods.