A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations

TL;DR

This paper introduces a high-order, energy-stable fully discrete Galerkin scheme for advection-diffusion equations, combining SBP and SAT techniques to achieve super-convergence and computational efficiency on coarse meshes.

Contribution

It develops a novel fully discrete SBP-SAT Galerkin framework that ensures stability and high accuracy for IBVPs in advection-diffusion problems.

Findings

Achieves super-convergence of order p+2 for p≥2.

Demonstrates energy stability through SBP-SAT formulation.

Efficiently captures space-time variations on coarse meshes.

Abstract

We present a high-order accurate fully discrete numerical scheme for solving Initial Boundary Value Problems (IBVPs) within the Continuous Galerkin (CG)-based Finite Element framework. Both the spatial and time approximation in Summation-By-Parts (SBP) form are considered here. The initial and boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) technique. The resulting SBP-SAT formulation yields an energy estimate in terms of the initial and external boundary data, leading to an energy-stable discretization in both space and time. The proposed method is evaluated numerically using the Method of Manufactured Solutions (MMS). The scheme achieves super-convergence in both spatial and temporal direction with accuracy $\mathcal{O}(p+2)$ for $p\geq 2$, where $p$ refers to the degree of the Lagrange basis. In an application case, we show that the fully discrete formulation efficiently captures space-time variations even on coarse meshes, demonstrating the method's computational effectiveness.