ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 28th Nov 2022, 08:24:28pm CET

Session Overview 
Session  
MS 8a: high order methods for nonlinear waves
 
Session Abstract  
Nonlinear waves are ubiquitous in nature, such as water waves in ocean dynamics, matter waves in quantum physics, soliton waves in optical fibers, etc. and many of them can be described by PDEs such as nonlinear Scrödinger equation, KleinGordon equation, Ko rtewegde Vries equation, Maxwell equation. Efficient and accurate numerical simulations for nonlinear wave PDEs are highly demanding in scientific and engineering computations. Spectral methods are capable of producing very accurate simulation results, and more im portantly, they require a substantially smaller number of unknowns (even for engineering accuracy) when compared to their lowerorder counterparts. Thus, spectral methods are especially attractive for emerging scientific applications with stringent accuracy and/or memory requirements. However, applications of spectral methods to nonlinear problems may vary quite differently depending on the setup of the nonlinear problems. Proper treatments have to be taken for different kinds of nonlinearities, to ensure the stability and accuracy as well as the numerical efficiency. This minisymposium aims at bringing together numerical analysts and computational sci entists to present their findings on recent advances in algorithm development and analysis of spectral methods for nonlinear wave problems. The topics will span the novel develop ments of spectral method and applications of spectral methods in solving nonlinear wave problems.  
Presentations  
12:00pm  12:30pm
EXACT SOLUTIONS OF THE WAVE EQUATION ROTATING ABOUT AN AXIS OF A SPHERE ^{1}Università di Modena e Reggio Emilia, Italy; ^{2}Università di Camerino, Italy We present a set of 3D eigenfunctions defined on the sphere, which can be used to build solutions of the wave equation (both in scalar of vector form) that rotate about the axis passing through the poles. The analytic expression of these solutions requires Bessel’s functions, as well as a set of orthogonal functions that are in relation with the Associated Legendre polynomials. Applications may be considered in the field of geophysics or, more in general, in astronomy. As an example, we show how this theory can be used to model the plasma cells appearing on the surface of the Sun. 12:30pm  1:00pm
Highorder finite elements in tokamak freeboundary plasma equilibrium computations Universite' Cote d'Azur, France, France The way a magnetic field influences the transport properties of charged particles is of high interest for a wide spectrum of physical systems and areas. It is a complex challenging problem [2] that goes beyond the purpose of the present paper. However, its suggests some key ingredients that need to be correctly treated from the mathematical and numerical points of view. The numerical simulation of the equilibrium of the plasma in a tokamak should benefit from higher regularity of the approximation of the magnetic flux map. We propose a finite element approach on a triangular mesh of the poloidal section, that couples piecewise linear finite elements in a region that does not contain the plasma and reduced HsiehCloughTocher [3] finite elements elsewhere. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details in the rest of the computational domain and simplifying the inclusion of ferromagnetic parts. The continuity of the numerical solution at the coupling interface is weakly enforced by mortar projection [1]. In this work we extend the method presented in [4] to the configuration where nonlinear ferromagnetic materials occupy the exterior part of the tokamak as is the case for WEST and JET tokamaks. Numerical simulations are performed with the code NICE described in [5]. References [1] C. Bernardi, Y. Maday, A.T. Patera. A new nonconforming approach to domain decomposition: The mortar element method. In Nonlinear Partial Differential Equations and Their Applications. in H. Brzis and J.L. Lions (eds.), Collge de France Seminar XI., 1992. [2] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics with Appli cations to Tokamaks. Series in Modern Applied Mathematics. Wiley GauthierVillars, Paris, 1989. [3] R.W. Clough, J.L. Tocher. Finite element stiffness matrices for analysis of plates in bending. In Proc. Conf. Matrix Methods in Struct. Mech., Air Force Inst of Tech., Wright Patterson A.F Base, Ohio, October 1965. [4] A. Elarif, B, Faugeras, F. Rapetti. Tokamak freeboundary plasma equilibrium computation using finite elements of class C 0 and C 1 within a mortar element approach. RR9364, INRIA Sophia Antipolis. 2020. hal02955007 [5] B. Faugeras, An overview of the numerical methods for tokamak plasma equilibrium computation implemented in the NICE code. Fusion Eng. Design, 160:112020, 2020. 1:00pm  1:30pm
Arbitrarily highorder exponential cutoff methods for preserving maximum principle of parabolic equations Southern University of Science and Technology, China, People's Republic of A new class of highorder maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear AllenCahn equation. The proposed method consists of a $k$thorder multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise $r$thorder polynomials and GaussLobatto quadrature. At every time level, the extra values violating the maximum principle are eliminated at the finite element nodal points by a cutoff operation. The remaining values at the nodal points satisfy the maximum principle and are proved to be convergent with an error bound of $O(\tau^k+h^r)$. The accuracy can be made arbitrarily highorder by choosing large $k$ and $r$. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phasefield problems. 1:30pm  2:00pm
on regularization methods for a backward timefractional diffusionwave problem ^{1}Bordeaux INP, France; ^{2}Xiamen University, School of mathematical sciences, China In this paper, we investigate numerical methods for a backward problem of the time fractional wave equation in bounded domains. We propose two fractional filter regulariza tion methods, which can be regarded as an extension of the classical Landweber iteration for the timefractional wave backward problem. The idea is first to transform the illposed backward problem into a weighted normal operator equation, then construct the regular ization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regu larized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the wellknown oversmoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity. 
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