Splitting methods: basics, analysis, modifications, and applications
数学专题报告
报告题目(Title)：Splitting methods: basics, analysis, modifications, and applications
报告人(Speaker)：Professor Alexander Ostermann（University of Innsbruck, Austria）
地点(Place)：后主楼1124
时间(Time)：9月7日（星期四），4:00pm-5:00pm
邀请人(Inviter)：蔡勇勇
报告摘要
Splitting methods are a well-established tool for the numerical integration of time-dependent partial differential equations. The basic idea behind these methods is to split the vector field into disjoint components, integrate them separately with an appropriate time step, and combine the individual flows to obtain the desired numerical approximation. Splitting methods play also a fundamental role in dynamic low-rank integrators.
The merits and pitfalls of splitting methods will be discussed on the basis of several examples. These include reaction-diffusion equations, the cubic nonlinear Schrödinger equation, the Vlasov-Poisson equations (a kinetic model in plasma physics), the Korteweg-de Vries equation, and the Kadomtsev-Petviashvili equation. It is shown that splitting methods can have superior geometric properties (such as preservation of positivity and favorable long term behavior) compared to standard time integration schemes. Furthermore, it is often possible to overcome a CFL condition that occurs in standard discretizations. Another advantage of splitting methods is that they can be implemented by making use of existing methods and codes for simpler problems, and that they often allow for parallelism in a straightforward way.
On the other hand, the use of splitting methods also requires a certain amount of care. The presence of (non-trivial) boundary conditions can lead to a strong order reduction and thus to computational inefficiency. Furthermore, non-smooth initial data pose a serious problem. For example, the traditional second-order Strang splitting requires four additional derivatives to solve the cubic nonlinear Schrödinger equation. To overcome all these problems, the integrators must be adapted accordingly as explained in the talk.
