Julie PATELA
Julie Patela is a Research Engineer and holds a PhD in Applied Mathematics. She is currently member of an R&D team at CEA whose objective is to develop new models and new numerical methods for HPC numerical simulation codes.
Journal of Computational Physics, Volume 518, 2024, 113325, ISSN 0021-9991, 2024
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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.
Communications in Computational Physics, 2023
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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
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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.
Thèse de Doctorat de l'Université Paris Cité, 2023
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The objective of this thesis is the development and the analysis of robust and accurate finite volume schemes for the approximation of the solution of the diffusion equation on deformed meshes with diffusion coefficient which can be anisotropic and/or discontinuous. To satisfy these properties, our schemes must preserve the positivity and achieve high-order accuracy. In this manuscript, we propose the first positivity-preserving arbitrary-order scheme for diffusion. Our approach is first to study the problem in 1D. In such a case, the positivity problem only appears for order 3 and higher. The 1D setting allows us to perform the mathematical analysis of this problem, including a proof of convergence of the scheme to an arbitrary order under a stability assumption. We then extend it to 2D at order 2, relying on well-known schemes. We study two possibilities: a DDFV-type scheme (Discrete Duality Finite Volume), which we compare with a method using polynomial reconstruction. Finally, this allows us to develop a monotonic scheme of arbitrary order on any mesh with a kappa diffusion coefficient that can be discontinuous and/or anisotropic. Improving the order is achieved through polynomial reconstruction, and monotonicity is obtained by reducing to a M-matrix structure, which gives nonlinear schemes. Each scheme is validated by numerical simulations showing the order of convergence and the positivity of the solution obtained.