Abstract : The numerical simulation of many physical systems requires an understanding of solutions which goes beyond mere numerical accuracy - they possess invariant structure and qualitative features that distinguish physically meaningful from spurious behaviour. This leads to the fundamental question to what extend such features of the underlying system can be preserved in the numerical flow, in particular over times much longer than is guaranteed by local error estimates. The proposed minisymposium brings together experts from computational mathematics to provide an overview of current state-of-the-art and recent advances in the study and design of methods for evolution equations with favourable long-time behaviour.
Organizer(s) : Yue Feng, Georg Maierhofer, Katharina Schratz
[03792] Bourgain techniques for error estimates at low regularity
Format : Talk at Waseda University
Author(s) :
Alexander Ostermann (Universität Innsbruck)
Lun Ji (Universität Innsbruck)
Frédéric Rousset (Université Paris-Saclay)
Katharina Schratz (Sorbonne Université)
Abstract : Standard numerical integrators suffer from order reduction when applied to nonlinear dispersive equations with non-smooth initial data. For such problems, we present filtered integrators that exhibit superior convergence rates at low regularity. Furthermore, due to the nonexistence of suitable embedding results, the error analysis at very low regularity cannot be carried out in standard Sobolev spaces. Instead, new techniques are required. They are based on Bourgain’s seminal work and will be sketched in the talk.
[01832] Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schr\"odinger equation
Format : Talk at Waseda University
Author(s) :
Yue Feng (Laboratoire Jacques-Louis Lions)
Abstract : In this talk, I will present the improved uniform error bounds of the time-splitting Fourier pseudospectral methods for the long-time dynamics of the Schr\"odinger equation with small potential and the nonlinear Schr\"odinger equation with weak nonlinearity. The main technique introduced is the regularity compensation oscillation (RCO), which control the high frequency modes by the regularity of the exact solution and the low frequency modes by phase cancellation and energy method.
[04274] A symmetric low-regularity approximation to the nonlinear Schrödinger equation
Format : Talk at Waseda University
Author(s) :
Yvonne Alama Bronsard (Sorbonne Université, LJLL)
Abstract : In this talk we will discuss the approximation to nonlinear dispersive equations which ask for low-regularity assumptions on the initial data, both for deterministic and random initial data.
We will put forth a novel time discretization to the nonlinear Schrödinger equation which allows for low-regularity approximation while maintaining good long-time preservation of the density and energy on the discrete level
[04742] Symmetric low regularity integrators via a forest formula
Format : Talk at Waseda University
Author(s) :
Yvain Bruned (Université de Lorraine)
Abstract : In this work, we will present general low regularity schemes that should encompass some of the symmetries of a given dispersive PDEs. We extend the low resonance decorated trees approach to a richer framework where we explore different ways of iterating Duhamel'formula and interpolating the lower part of the resonance for a Taylor approximation. This gives more degrees of freedom, encapsulated via a new forest formula that provides the general form for these new schemes. From this formula, we are able to derive conditions on the coefficients for finding new symmetric schemes. We believe that such an approach could tackle other symmetries.
[04423] Quantum computation of partial differential equations
Format : Talk at Waseda University
Author(s) :
Shi Jin (Shanghai Jiao Tong University)
Abstract : Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to technology revolution in the 21st century. Since quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. The most efficient quantum PDE solver is quantum simulation based on solving the Schrodinger equation. It became challenging for general PDEs, more so for nonlinear ones. Our talk will cover two topics:
1) We introduce the “warped phase transform” to map general linear PDEs and ODEs to Schrodinger equation or with unitary evolution operators in higher dimension so they are suitable for quantum simulation;
2) For (nonlinear) Hamilton-Jacobi equation and scalar nonlinear hyperbolic equations we use the level set method to map them—exactly—to phase space linear PDEs so they can be implemented with quantum algorithms and we gain quantum advantages for various physical and numerical parameters.
[05333] Asymptotic expansions for the linear PDEs with oscillatory input terms: Analytical form and error analysis
Format : Talk at Waseda University
Author(s) :
Karolina Joanna Kropielnicka (Institute of Mathematics of Polish Academy of Sciences)
Abstract : Partial differential equations with highly oscillatory input term are hardly ever solvable analytically and they are difficult to treat numerically. Modulated Fourier expansion used as an ansatz is a well known and extensively investigated tool in asymptotic numerical approach for this kind of problems.
In this talk I will consider input term with single frequency and will show that the ansatz need not be assumed – it can be derived naturally while developing formulas for expansion coefficients. Moreover I will present the formula describing the error term and its estimates. Theoretical investigations will be illustrated by results of the computational simulations.
[03756] A new picture on the Strang Splitting
Format : Talk at Waseda University
Author(s) :
Juan Del Valle (University of Gdansk)
Karolina Kropielnicka (Polish Academy of Sciences)
Abstract : Strang splitting is a well-established and widely used technique for finding approximate solutions of linear differential equations of the type u’=(A+B)u, where A and B are time-independent components. However, it can also be used for the case of time-dependent component B(t) after the application of the mid-point quadrature rule at the level of the Magnus expansion. However, the error estimate is absent in the case of singular cases of unbounded operators B(t).
In this talk, I will show how Strang splitting scheme for time-dependent components can be derived using the Duhamel formula. Based on this approach, I will (i) present a new proof of convergence of this scheme and (ii) elaborate on the possibilities brought by this approach for higher order methods. A concrete analysis of the error estimated and numerical simulations will be presented for the physically relevant example of a hydrogen atom featuring the singular Coulomb potential.
[05308] The role of breathers in the formation of extreme ocean waves
Format : Talk at Waseda University
Author(s) :
Amin Chabchoub (Kyoto University )
Abstract : The modulation instability is a fundamental mechanism, which explains localized wave focusing processes in dispersive wave systems. When considering the nonlinear Schrödinger equation as underlying wave model, exact breather solutions are particularly useful to initiate and control unstable wave dynamics in a numerical or laboratory experiment. This talk will summarize the main experimental achievements on breathers and connect these findings to the dynamics of ocean rogue waves.
[04908] Structure-preserving finite element discretization of nonlinear PDEs
Format : Talk at Waseda University
Author(s) :
Ari Stern (Washington University in St. Louis)
Abstract : This talk discusses some recent advances in structure-preserving methods for nonlinear PDEs, combining finite elements in space and geometric integration in time. In particular, we extend some earlier methods and results to a broader class of Hamiltonian PDEs than previously considered, showing that multisymplectic and other conservation laws are preserved. These methods apply on unstructured meshes, not just structured grids, and may be arbitrarily high-order.
[01990] High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation
Format : Talk at Waseda University
Author(s) :
Genming Bai (The Hong Kong Polytechnic University)
Jiashun Hu (The Hong Kong Polytechnic University)
Buyang Li (The Hong Kong Polytechnic University)
Abstract : A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schr\"odinger equation based on Gauss collocation in time and finite element discretization in space, by introducing a mass- and energy-correction post-process at every time level. The existence, uniqueness and high-order convergence of the numerical solutions are proved. In particular, the error of the numerical solution is $O(\tau^{k+1}+h^p)$ in the $L^\infty(0,T;H^1)$ norm after incorporating the accumulation errors arising from the post-processing correction procedure at all time levels, where $k$ and $p$ denote the degrees of finite elements in time and space, respectively, which can be arbitrarily large. Several numerical examples are provided to illustrate the performance of the proposed new method, including the conservation of mass and energy, and the high-order convergence in simulating solitons and bi-solitons.
[03931] Geometric two-scale integrators for highly oscillatory system
Format : Talk at Waseda University
Author(s) :
Bin Wang (Xi'an Jiaotong University)
Abstract : In this talk, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy at finite times and some near-conservation laws in long times.
[03985] Structure preserving schemes for Allen--Cahn type equations
Format : Talk at Waseda University
Author(s) :
Yongyong Cai (Beijing Normal University)
Abstract : In comparison with the Cahn--Hilliard equation, the classic Allen--Cahn equation satisfies the maximum bound principle (MBP) but fails to conserve the mass. Here, we report the MBP and corresponding numerical schemes for the Allen--Cahn equation with nonlocal constraint for the mass conservation. As an extension, we discuss the case of the convective Allen--Cahn equation.
