A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914241338933248 |
|---|---|
| author | Mandal, Mrityunjoy Nordström, Jan Malan, Arnaud G |
| author_facet | Mandal, Mrityunjoy Nordström, Jan Malan, Arnaud G |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05071 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations Mandal, Mrityunjoy Nordström, Jan Malan, Arnaud G Mathematical Physics Numerical Analysis 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. |
| title | A high order accurate and provably stable fully discrete continuous Galerkin framework on summation-by-parts form for advection-diffusion equations |
| topic | Mathematical Physics Numerical Analysis |
| url | https://arxiv.org/abs/2601.05071 |