# Numerics of PDEs unplugged 2020

In summer 2017 a workshop called PDEs_unplugged∂Karlsruhe took place at KIT. Unplugged means that only blackboard talks were given, and that mobile phones and laptops were switched off during all talks. This very special format was highly appreciated by all participants. The intention of Numerics of PDEs unplugged 2020 is to transfer this concept to numerical analysis. This is quite paradoxical considering that this field of mathematics is actually motivated by the use of computers. Is it possible to give talk on numerical methods for PDEs on a blackboard, and without running movies or showing convergence plots? We believe: it is! And we believe that the result will often be much better than so many conference talks with a speaker clicking through 60 slides in 25 minutes.

Sadly, the Covid-19 pandemia put a spoke in our wheel. The Corona restrictions forced us – irony of fate – to turn Numerics of PDEs unplugged 2020 into a pure online event. All talks will be given as a video conference via zoom, but our speakers will try to preserve as much as possible of the original unplugged spirit.

## Registration

When you want to attend this online event please register. All you need to do so is composing an email to admin∂waves.kit.edu including your full name and your affiliation.

The details to the zoom meeting will be sent via email (not later than) on October 29th noon (UTC +1).

## Date and place

October 29-30, 2020
@KIT · Karlsruhe
online using zoom

## Speakers

 Lehel Banjai · Edinburgh (Scotland) David Cohen · Gothenburg (Sweden) Balázs Kovács · Regensburg (Germany) Caroline Lasser · Munich (Germany) María López-Fernández · Málaga (Spain) Christian Lubich · Tübingen (Germany) Alexander Ostermann · Innsbruck (Austria) Gilles Vilmart · Geneva (Switzerland)

## Scientific program

Thursday, October 29th
14:00 Caroline Lasser · Various variational principles
Abstract: We discuss different variational principles, that have been used in the chemical physics' literature for solving the time-dependent Schrödinger equation, together with simple a posteriori error estimates that allow for mimimal computational overhead.
This is joint work with Chunmei Su.
15:00 Lehel Banjai · FEM/BEM coupling for the wave equation in the second order formulation
Abstract: We consider acoustic scattering of waves by bounded inhomogeneities in an unbounded homogeneous domain. We make use of a symmetric FEM/BEM formulation. The solution in the interior is discretized in space by a finite element method. In the unbounded outer domain, the solution is represented by layer potentials and discretized on the boundary by the boundary element method. In time we discretize the boundary integral operators by convolution quadrature based on the trapezoidal rule, whereas the interior equation is discretized by leapfrog under a CFL condition.
We give a stability and convergence analysis, and discuss connections to existing literature.
16:15 David Cohen · Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations
Abstract: We analyze the qualitative properties and the orders of convergence of a splitting scheme for a class of nonlinear stochastic  Schrödinger equations driven by additive Itô noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities.
This is based on a joint work with Charles-Edouard Bréhier, Université Claude Bernard Lyon 1.
17:15 Gilles Vilmart · Convergence analysis of explicit stabilized integrators for parabolic semilinear (stochastic) PDEs
Abstract: Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff dissipative problems.
In this "unplugged talk", we present a convergence analysis of a family of explicit stabilized schemes applies to classes of deterministic and stochastic semilinear diffusion PDEs.
Joint work with Assyr Abdulle (EPF Lausanne) and Charles-Edouard Bréhier (Univ Lyon).
Friday, October 30th
09:00 Alexander Ostermann · Time integration at low regularity
Abstract: Standard numerical integrators such as splitting methods or exponential integrators suffer from order reduction when applied to semi-linear dispersive problems with non-smooth initial data. In this talk, we focus on nonlinear Schrödinger equations. For such problems, we present and analyze a class of (filtered) Fourier integrators that exhibit (superior) convergence rates at low regularity.
This is joint work with K. Schratz (Sorbonne U, Paris), F. Rousset (U Paris-Sud) and M. Knöller (KIT).
10:00 Balázs Kovács · A convergent algorithm for Willmore flow of closed surfaces -- with analog numerical experiments
Abstract: We will sketch a proof of convergence is for semi-discretisations of Willmore flow of closed two-dimensional surfaces. The proposed method discretizes a system coupling evolution equations along the flow for the normal vector and mean curvature with the velocity law. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm semi-discrete error bounds. The main focus of the talk will be on the stability analysis. The stability proof uses a non-standard version of energy estimates, testing with both the error and its time derivative.
We will (at least, will attempt) to present some „analog“' numerical experiments as well.
The talk is based on joint work with Buyang Li and Christian Lubich.
11:15 Christian Lubich · Analysis of Strang splitting for the semiclassical Schrödinger equation
Abstract: For the semiclassically scaled time-dependent Schrödinger equation, which describes, e.g. quantum dynamics of nuclei in a molecule, we study time discretization by splitting the Hamiltonian operator into its potential and kinetic energy parts. We briefly review the $L^2$ error bound by Descombes and Thalhammer (2010), which is an $O(\Delta t^2 / \epsilon)$ error bound, where $\epsilon$ is the small semiclassical parameter. The main part of the talk is about an $O(\Delta t^2 + \epsilon^2)$ error bound for expectation values of observables. The proof plugs basic results from semiclassical analysis on the familiar $O(\Delta t^2)$ error bound for the Störmer-Verlet method applied to the corresponding classical equations of motion. The result appears as Theorem 7.4 in the Acta Numerica 2020 article written jointly with Caroline Lasser, "Computing quantum dynamics in the semiclassical regime" (arXiv:2002.00624).
12:15 Unfortunately, María López-Fernández had to cancel her participation.
María López-Fernández · Pseudospectral roaming contour integral methods for convection-diffusion equations
Abstract: We propose a general framework to apply different types of contour integral methods to convection-diffusion PDEs and in particular those arising in finance. These methods aim to determine a numerical approximation of the solution by computing the inverse Laplace transform associated to the problem. Our approach relies on a new fast pseudospectral roaming technique. We optimize the integration contours for a given problem, target accuracy and time window of interest. Elliptic, parabolic and hyperbolic profiles proposed in the literature are investigated and compared.
The talk is based on collaboration with Nicola Guglielmi, Giancarlo Nino and Mattia Manucci.

The talks are scheduled to be of 50 minutes length (including discussion).

## Local organizers

• Marlis Hochbruck
• Tobias Jahnke