European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   Time Integration and Spatial Discretisations (Part 2)
Chair: Francisco Gaspar
10:40
25 mins
A conservative compact finite difference scheme for the coupled Schr\"{o}dinger-KdV equations
Shusen Xie, Su-Cheol Yi
Abstract: A conservative compact finite difference scheme is presented to numerically solve the coupled Schr\"{o}dinger-KdV equations. The analytic solutions of the coupled equations have some invariants such as the number of plasmons, the number of particles and the energy of oscillations etc., and we proved that the compact difference scheme preserves those invariants in discrete sense. Optimal order convergence rate of the proposed compact scheme was analyzed. Numerical experiments on model problems show that the scheme is of high accuracy.
11:05
25 mins
Semi implicit relaxation schemes for low-mach flows
Emmanuel Franck, Laurent Navoret, Françcois Bouchut
Abstract: In this work we consider the time discretization of compressible fluid models which appear in gas dynamics, biology, astrophysics or plasma physics for Tokamaks. In general, for the hyperbolic sys- tem we use an explicit scheme in time. However, for some applications, the characteristic velocity of the fluid is very small compared to the fastest velocity speed. For the Euler equations we speak about low-Mach regime. In this case, to filter the fast scales it is common to use an implicit scheme. The implicit schemes allow to filter the fast scales that we don’t want to consider and choose a time step independent of the mesh step and adapted to the characteristic velocity of the fluid. The problem induced by the discretization of the hyperbolic system are nonlinear and ill-conditioned in the regime considered. In this work we propose an alternative method based on the relaxation schemes. This method allows to approximate the full complex nonlinear problem by a problem larger, but simpler (more or totally linear) with a stiff source term. Apply a splitting between the source and the rest of the model allows to design a simple scheme after. After have introduced the general method, we propose study different relaxation choices and design for each choice a semi implicit/implicit time scheme simpler that the original one. After that we will discuss the property of the time schemes and the spatial discretization. At the end we will show that with the good choice of relaxation, we obtain a scheme, valid for all the regimes, very efficient in the low mach regime with a very simple implicit part. We will compare different schemes on some numerical tests in 1D/2D. To finish we will discuss the extension at the MHD and the Euler gravity models.
11:30
25 mins
Conservative numerical schemes for nonlinear Schrödinger equations in complex physical setups
Johan Wärnegård, Patrick Henning
Abstract: In this talk we compare various mass-conservative time-integrators for the Gross-Pitaevskii equation in physically relevant setups. The comparison contains methods that are purely mass-conservative, methods that are additionally symplectic and methods that preserve the energy exactly. Our finding is that in particular in low-regularity regimes, e.g. in the context of rapidly oscillating potentials or close to quantum phase transitions, the numerical approximations are very sensitive to small variations in the discrete energy. Consequently, mass conservation alone will not lead to a competitive method in complex settings. More notably, the differences between symplectic and energy-conservative discretizations turn out to be stronger than expected if the regularity of the solution is low. The energy-conservative schemes achieve a visibly higher accuracy in our test cases. The talk concludes with a simulation of the phase transition from a (pseudo) Mott insulator to a superfluid.