Julie PATELA
Ingénieur - Chercheur et titulaire d’un doctorat en Mathématiques Appliquées, Julie Patela travaille au sein d’une équipe de R&D du CEA dont l’objectif est de développer de nouveaux modèles et de nouvelles méthodes numériques pour les codes de simulation numérique HPC.
Communications in Computational Physics, 2023
abstract
Abstract
The DDFV (Discrete Duality Finite Volume) method is a finite volume scheme mainly dedicated to diffusion problems, with some outstanding properties. This scheme has been found to be one of the most accurate finite volume methods for diffusion problems. In the present paper, we propose a new monotonic extension of DDFV, which can handle discontinuous tensorial diffusion coefficient. Moreover, we compare its performance to a diamond type method with an original interpolation method relying on polynomial reconstructions. Monotonicity is achieved by adapting the method of Gao et al [A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes] to our schemes. Such a technique does not require the positiveness of the secondary unknowns. We show that the two new methods are second-order accurate and are indeed monotonic on some challenging benchmarks as a Fokker-Planck problem.
Computational & Applied Mathematics, vol 42, 2023
abstract
Abstract
When solving numerically an elliptic problem, it is important in most applications that the scheme used preserves the positivity of the solution. When using finite volume schemes on deformed meshes, the question has been solved rather recently. Such schemes are usually (at most) second-order convergent, and non-linear. On the other hand, many high-order schemes have been proposed that do not ensure positivity of the solution. In this paper, we propose a very high-order monotonic (that is, positivity preserving) numerical method for elliptic problems in 1D. We prove that this method converges to an arbitrary order (under reasonable assumptions on the mesh) and is indeed monotonic. We also show how to handle discontinuous sources or diffusion coefficients, while keeping the order of convergence. We assess the new scheme, on several test problems, with arbitrary (regular, distorted, and random) meshes.
Abstract
Monotonicity is very important in most applications solving elliptic problems. Many schemes preserving positivity has been proposed but are at most second-order convergent. Besides, in general, high-order schemes do not preserve positivity. In the present paper, we propose an arbitrary-order monotonic method for elliptic problems in 2D. We show how to adapt our method to the case of a discontinuous and/or tensorvalued diffusion coefficient, while keeping the order of convergence. We assess the new scheme on several test problems.