Mini- course: Modern Time-Stepping Techniques for PDEs: Splitting and Low Regularity Approaches

Mini- course: Modern Time-Stepping Techniques for PDEs: Splitting and Low Regularity Approaches

Time:08:00:25/07/2025 to  16:30:29/07/2025

Venue/Location: Vietnam Institute for Advanced Study in Mathematics (VIASM), 161 Huynh Thuc Khang, Hanoi

Content:

Mini-course 1: Splitting Methods: Basics, Challenges, and Applications

Splitting methods are tools that efficiently solve differential equations by breaking them down into smaller parts. The basic idea is to divide the vector field of the differential equation into separate components, integrate each of them separately, and then combine the results after each time step. This is a simple and often efficient procedure. However, splitting methods require a careful handling of non-trivial boundary conditions. Adaptations to the integrators are necessary to address this issue. These lectures will cover the construction and numerical analysis of splitting methods, typical applications, order reduction due to nontrivial boundary conditions, and the efficient combination of splitting with exponential integration in sonic boom calculations.

  1. Splitting methods for PDEs
  2. Typical applications (Vlasov, KdV, KP)
  3. Boundary corrections
  4. Sonic boom computations 

Mini-Course  2: Low Regularity Time Integration of Dispersive Problems

Standard numerical integrators, such as Lie splitting, Strang splitting, and exponential integrators, experience order reduction when applied to semilinear dispersive problems with non-smooth initial data. To address this issue, a recent development introduces a new class of integrators known as low-regularity integrators. These integrators use the variation-of-constants formula and employ resonance-based approximations in Fourier space, demonstrating improved convergence rates at low regularity. However, the estimation of nonlinear terms in the global error still relies on classical bilinear estimates derived from Sobolev embeddings. At very low regularity, traditional error analysis in Sobolev spaces is hampered by the lack of suitable embeddings. A novel framework, inspired by Bourgain's techniques, has been developed that allows the analysis of methods applicable to very low regularity initial data. This approach has been applied to various problems, including the nonlinear Schrödinger equation and the `good' Boussinesq equation.

  1. Fourier integrators – part 1
  2. Fourier integrators – part 2
  3. Bourgain techniques – part 1
  4. Bourgain techniques – part 2

Organizing Committee 

  • Le Minh Ha, VIASM
  • Vu Thai Luan, Texas Tech University, USA

Lecturer: Alexander Ostermann, University of Innsbruck, Austria

Expected participants: Advanced undergraduate students, graduate students, lecturers/researchers

Format: Hybrid

Note: Online participation is only available for attendees outside of Hanoi.

Registration:Please register via link: https://forms.gle/vqwR74XgQ4mJGrHj8

Deadline for registration: July 20, 2025

Contact: Secretary Le Hung Hai (VIASM), Email: lhhai@viasm.edu.vn