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).

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Session Overview
MS 8b: high order methods for nonlinear waves
Friday, 16/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

SAV approaches for nonlinear Hamiltonian systems

Jie Shen

Purdue University, United States of America

I shall present efficient SAV approaches and their error analysis for a class of nonlinear Hamiltonian systems, including nonlinear Schrodinger equations, BECs, Klein-Gordon equations. I shall also discuss how to use Lagrange multiplier SAV approach to construct efficient schemes which can conserve multiple Hamiltonians.

12:30pm - 1:00pm

Multiscale methods and analysis for the Dirac equation in the nonrelativistic regime

Weizhu Bao

National University of Singapore, Singapore

In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac equation in the nonrelativistic regime. This is a joint work with Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin.

1:00pm - 1:30pm

Efficient Spectral Methods for Fractional PDEs with Applications to Electromagnetic Waves in Dispersive Media

Li-Lian Wang

Nanyang Technological University, Singapore

In this talk, we shall present efficient spectral methods using non-tensorial generalised Hermite functions and mapped Chebyshev functions for integral fractional Laplacian in multiple dimensions, and discuss some fractional PDEs related to electromagnetic waves in dispersive media. More precisely, we shall introduce two types of non-standard basis functions that can diagonalise the integral fractional Laplacian in R^d and lead to fast and accurate spectral algorithms for this global operator and also the Schrodinger operator with fractional power potentials.

1:30pm - 2:00pm

Uniformly accurate NPI methods for nonlinear Klein-Gordon equations in the nonrelativistic regime

Yongyong Cai

Beijing Normal Unnivesity, China, People's Republic of

A class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods is proposed for the nonlinear Klein-Gordon equation (NLKG) in the nonrelativistic regime, involving a dimensionless parameter $\varepsilon\ll1$ inversely proportional to the speed of light. For $0<\varepsilon\ll1$, the solution propagates waves in time with $O(\varepsilon^2)$ wavelength, which brings significant difficulty in designing accurate and efficient numerical schemes.

Detailed constructions of the NPI methods up to the third order in time are demonstrated for NLKG with a cubic/quadratic nonlinear term.

In addition, the implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly shown, and the corresponding error estimates are rigorously analyzed. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes. This is a joint work with Xuanxuan Zhou.

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