MB2: Recent Advances in Time-Integration of PDEs II

Room: Old Main Academic Center 3050

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OrganizerVu Thai Luan, Mississippi State University

LiLi Ju, University of South Carolina

Time: 10:00 am - 10:25 am (CST)

Title: Stabilized Integrating Factor Runge-Kutta Method and Unconditional Preservation of Maximum Bound Principle

Abstract: Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that their solution preserves for all time a uniform pointwise bound in absolute value imposed by the initial and boundary conditions. It has been a challenging problem on how to design unconditionally MBP-preserving time stepping schemes for these equations, especially the ones with order greater than one. We combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive the sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many strong stability preserving sIFRK (SSP-sIFRK) schemes do not satisfy these conditions, except the first-order one. Various numerical experiments are also carried out to demonstrate the performance of the proposed method.

James Lambers, University of Southern Mississippi

Time: 10:25 am - 10:50 am (CST)

Title: A Spectral Multistep Method for Time-Dependent PDEs

Abstract: Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for partial differential equations (PDEs) with stability characteristic of implicit methods. Unlike other time-stepping approaches, KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As a result, KSS methods scale effectively to higher spatial resolution. This talk will present a multistep formulation of KSS methods to provide a "best-of-both-worlds” situation that combines the efficiency of multistep methods with the stability and scalability of KSS methods. The effectiveness of spectral multistep methods will be demonstrated using numerical experiments.

Joint work with Bailey Rester.

Thi Thao Phuong Hoang, Auburn University

Time: 10:50 am - 11:15 am (CST)

Title: Low Regularity Integrators for Semilinear Parabolic Equations with Maximum Bound Principles

Abstract: This talk is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.

This is joint work with Cao-Kha Doan (Auburn University), Lili Ju (University of South Carolina), and Katharina Schratz (Sorbonne Université).

Rujeko Chinomona, Temple University

Time: 11:15 am - 11:40 am (CST)

Title: Derivation of Stability Optimized IMEX-MRI-GARK Methods

Abstract: Towards accurate and efficient time integration of systems of differential equations that exhibit multiple time scales, multirate methods have been developed. Of particular interest are multirate methods of an infinitesimal nature which offer flexibility in the choice of integrator for the slow and fast dynamics. Despite a recent uptick in investigations into multirate methods, a well established stability theory is yet to be developed. In this talk we will focus on the derivation of implicit-explicit multirate infinitesimal generalized-structure additive Runge-Kutta (IMEX-MRI-GARK) methods that are stability optimized. We define a notion of joint stability based on analysis of the slow-explicit component. We demonstrate that methods with larger joint stability regions perform better than those with smaller regions.

Alex Fish, Southern Methodist University

Time: 11:40 am - 12:05 pm (CST)

Title: Adaptive Time Step Control for Infinitesimal Multirate Methods

Abstract: We extend the single-rate controller work of Gustafsson (1994) to the context of multirate methods, methods which temporally evolve initial-value problems by using different time step sizes for components of the problem which evolve on distinct time scales. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the ``fast'' and ``slow'' time scales within infinitesimal multirate (MRI) methods. We then investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. Finally, we numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. We combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.