To illustrate this work we propose to compare the exact solution diffusion (top, left) on Cartesian mesh, the numerical solution given by the classical scheme for hyperbolic heat equation (top, right) on Kershaw mesh (first figure), the numerical solution given by the edge asymptotic preserving scheme (bottom, left) on Kershaw mesh and the numerical solution given by the nodal asymptotic preserving (bottom, right) on Kershaw mesh.
The second way use a different point of view. We propose to design a hyperbolic scheme using a geometrically compatible with a convergent diffusion scheme correctly chooses. After we modify the hyperbolic scheme using the Jin-Levermore method to obtain an asymptotic preserving scheme. In this case we use the Breil-Maire [BM07] and MPFA [AM08] diffusion schemes.
Key words : Hyperbolic heat equation, unstructured meshes, finite volume methods, nodal scheme, AP schemes, hyperbolic system with source terms. |
Proceedings : An asymptotic preserving scheme for P1 model using classical diffusion schemes on unstructured polygonal meshes, E. Franck, P. Hoch, P. Navaro and G. Samba, ESAIM: PROCEEDINGS, October 2011, Vol. 32, p. 56-75. | |
Proceedings : A priori analysis of asymptotic preserving schemes with the modified equation. B. Després (principal autor), C. Buet et E. Franck. Hyperbolic problems: theory, numerics and applications, AIMS on Applied Mathematics,vol 8, pp 501. | |
Paper : Design of asymptotic preserving finite volume schemes for hyperbolic heat equation on unstructured meshes , C. Buet, B. Després, E. Franck, Numerische Mathematik, October 2012, Volume 122, Issue 2, pp 227-278. | |
Paper : Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes , C. Buet, B. Després, E. Franck, T. Leroy, Mathematics of computation, 12 Septembre 2016. |
Key words : Friedrichs system, discrete ordinate method, spherical harmonics expansion, unstructured meshes, finite volume methods, nodal scheme, AP schemes, hyperbolic system with source terms, micro-macro decomposition. |
Paper : Asymptotic preserving schemes for Friedrichs systems with stiff relaxation on unstructured meshes: applications to the angular discretization models in linear transport, C. Buet, B. Després, E. Franck, Journal Scientific Computing. |
Key words : radiative transfer, unstructured meshes, finite volume methods, non linear moments model, AP schemes, entropy, maximum principle. |
Proceedings : Asymptotic Preserving Finite Volumes Discretization For Non-Linear Moment Model On Unstructured Meshes, C. Buet, B. Després, E. Franck, Finite Volumes for Complex Applications VI Problems and Perspectives, Springer Proceedings in Mathematics Volume 4, 2011, pp 467-474. |
Paper : An asymptotic preserving scheme with the maximum principle for the M1 model on distorted meshes, C. Buet, B. Després, E. Franck, Comptes Rendus Mathematique, Volume 350, Issues 11-12, June 2012, Pages 633-638. |
Key words : Euler equations, unstructured meshes, finite volume methods, AP schemes, entropy, well-balanced methods, Lagrange+remap scheme. |
Proceedings : Modified Finite Volume Nodal Scheme for Euler Equations with Gravity and Friction, E. Franck, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects Springer Proceedings in Mathematics & Statistics Volume 77, 2014, pp 285-292. |
Preprint : Modified finite volume scheme with local high order discretization of hydrostatic equilibrium for Euler equations with external forces, E. Franck, L. Mendoza |
EUROfusion project research (2014): |
EUROfusion Enabling Research Project (2015-2017): Global non-linear MHD modeling in toroidal geometry of disruptions, edge localized modes, and techniques for their mitigation and suppression. Hoelzl M. (PI),Becoulet M., Sonnendruecker E., Strumberger E., Pautasso G., Ratnani A., Orain F., Nardon E., Dif-Pradalier G., Latu G., Grandgirard V., Passeron C., Morales J., Nkonga B., Guillard H., Sangam A., Franck E., Pamela S., Cahyna P., Seidl J., Futatani S., Westerhof E. |
Key words : nonlinear time schemes, preconditioning, stability, multiscale problems, reduced MHD, plasma physics, Jorek code |
Preprint : Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code, E. Franck, M. Hölzl, A. Lessig, E. Sonnendrücker. |