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

Mrityunjoy Mandal, Jan Nordström, Arnaud G Malan

TL;DR

Addresses the need for high-order, energy-stable, fully discrete CG discretizations for advection–diffusion IBVPs. The authors develop a space-time SBP–SAT framework with CG in space and SBP in time, incorporating multistage time integration and weak boundary conditions, and prove energy stability across continuous, semi-discrete, and fully discrete levels. They report super-convergence of order $\mathcal{O}(p+2)$ in space and time for $p\ge 2$ via MMS, and demonstrate efficiency on coarse space-time meshes in a representative application. The work offers a robust, accurate, and computationally efficient tool for simulating advection–diffusion phenomena across regimes, with potential extension to nonlinear problems.

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.

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

TL;DR

Addresses the need for high-order, energy-stable, fully discrete CG discretizations for advection–diffusion IBVPs. The authors develop a space-time SBP–SAT framework with CG in space and SBP in time, incorporating multistage time integration and weak boundary conditions, and prove energy stability across continuous, semi-discrete, and fully discrete levels. They report super-convergence of order in space and time for via MMS, and demonstrate efficiency on coarse space-time meshes in a representative application. The work offers a robust, accurate, and computationally efficient tool for simulating advection–diffusion phenomena across regimes, with potential extension to nonlinear problems.

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 for , where 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.
Paper Structure (17 sections, 28 equations, 3 figures, 5 tables)

This paper contains 17 sections, 28 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. Continuous energy analysis
  4. Weak imposition of boundary conditions for the continuous problem
  5. The semi-discrete approximation
  6. Constructions of the SBP operators at the element level for one-dimensional domain
  7. Construction of the global matrices
  8. The semi-discrete energy analysis
  9. The fully-discrete system
  10. The fully discrete energy analysis
  11. The multistage-based temporal formulation
  12. The fully discrete energy analysis for multistage based formulation
  13. Numerical results
  14. MMS study: temporal mesh convergence
  15. MMS study: spatial mesh convergence
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Convergence of solution at time $t=1$ sec for (left) $g=1$ and (right) $g(t)$ takes the form of (\ref{['Eq:InflowBC_timeVarying_g']})
  • Figure 2: Convergence of solution at $x=0.9$ for $g=1$ (left) and (right) $g(t)$ takes the form of (\ref{['Eq:InflowBC_timeVarying_g']})
  • Figure 3: Cost-efficiency studies for amplitude $\alpha=1$ with various frequencies (a) $\omega=5\pi$, (b) $\omega=10\pi$, and (c) $\omega=20\pi$

Theorems & Definitions (1)

  • Remark 6.1