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Session Overview
MS 8a: high order methods for nonlinear waves
Thursday, 15/July/2021:
12:00pm - 2:00pm

Session Chair: Yongyong Cai
Session Chair: Li-Lian Wang
Virtual location: Zoom 5

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, Klein-Gordon 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 lower-order 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 set-up 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 mini-symposium 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


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12:00pm - 12:30pm


Daniele Funaro1, Lorella Fatone2

1Università di Modena e Reggio Emilia, Italy; 2Università 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

High-order finite elements in tokamak free-boundary plasma equilibrium computations

Blaise Faugeras, Francesca Rapetti

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

piece-wise linear finite elements in a region that does not contain the plasma and reduced

Hsieh-Clough-Tocher [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].


[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 Gauthier-Villars,

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 free-boundary plasma equilibrium

computation using finite elements of class C 0 and C 1 within a mortar element approach.

RR-9364, INRIA Sophia Antipolis. 2020. hal-02955007

[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 high-order exponential cut-off methods for preserving maximum principle of parabolic equations

Jiang Yang

Southern University of Science and Technology, China, People's Republic of

A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a $k$th-order multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss--Lobatto quadrature. At every time level, the extra values violating the maximum principle are eliminated at the finite element nodal points by a cut-off 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 high-order 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 phase-field problems.

1:30pm - 2:00pm

on regularization methods for a backward time-fractional diffusion-wave problem

Majdi Azaiez1, Bin Fan2, Chuanju Xu2

1Bordeaux INP, France; 2Xiamen 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 time-fractional wave backward problem. The idea is first to transform the ill-posed

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 well-known over-smoothing

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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