MA2:  Recent Advances in Time-Integration of PDEs  I

Room: Old Main Academic Center 3050

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OrganizerVu Thai Luan, Mississippi State University

Alexander Ostermann, University of Innsbruck, Austria

Time: 10:00 am - 10:25 am (CST)

Title: Boundary Corrected Strang Splitting

Abstract: Strang splitting is a well-established method for the numerical integration of evolution equations. It allows parts of the vector field to be integrated using specially tailored methods and the numerical approximation to be represented as a composition of these (often simpler) flows. Such a splitting approach can lead to high computational efficiency. However, if the original problem is subject to non‐periodic boundary conditions, a simple splitting of the vector field can also lead to problems: the accuracy of the numerical solution suffers, and the observed order of convergence is reduced.

The reason for this behavior is some incompatibility of the intermediate steps in the splitting with the prescribed boundary conditions. One remedy is to modify the boundary conditions for the intermediate steps to avoid these incompatibilities. However, the exact form of this modification requires careful analysis of the problem under consideration. In this talk, I will explain some strategies that have been developed in recent years to address this problem.

This is a joint work with Lukas Einkemmer and Thi‐Tam Dang.

Marco Caliari, University of Verona, Italy

Time: 10:25 am - 10:50 am (CST)

Title: A μ-mode Approach for Exponential Integrators: Action of φ-functions of Kronecker sums

Abstract: Due to the importance of simulation in various fields of science and engineering, devising efficient computational methods for solving high-dimensional evolutionary Partial Differential Equations is of paramount importance. After semidiscretization in space, the arising system of Ordinary Differential Equations is often stiff and thus the time evolution requires adequate schemes, such as exponential integrators. To be competitive, these methods require the efficient approximation of the action of the so-called matrix φ-functions. In this talk, we present an innovative and efficient way to compute the just mentioned action when the ODEs system has a Kronecker structure, that is its linear part M∈Cnd×nd can be written as a Kronecker sum of matrices

Ad⊕···⊕A1, Aμ∈Cn×n,

being d and n the number of space dimensions and one-dimensional d.o.f., respectively. The approach only requires the computation of few exponentials of the small matrices Aμ and suitably combines them through the tensor operation known as μ-mode product. We conclude the talk by presenting some numerical examples in three space dimensions, for which


by using recently designed exponential Runge–Kutta methods implemented with our technique, in comparison with state-of-the-art algorithms for the computation of the action of linear combinations of matrixφ-functions.

This is joint work with Fabio Cassini and Franco Zivcovich.

Amnon J Meir, Southern Methodist University

Time: 10:50 am - 11:15 am (CST)

Title: On the Equations of Electroporoelasticity

Abstract: Complex coupled problems are mathematical models of physical systems(or physically motivated problems) which are governed by partial differential equations and which involve multiple components, complex physics or multi-physics, as well as complex or coupled domains, or multiple scales. One such phenomenon is electroporoelasticity. After introducing the equations of electroporoelasticity (the equations of poroelasticity coupled to Maxwell's equations) which have applications in geoscience, hydrology, and petroleum exploration, as well as various areas of science and technology, I will describe some recent results(well posedness), the numerical analysis of a finite-element based method for approximating solutions, and some challenges.

David Shirokoff, New Jersey Institute of Technology

Time: 11:15 am - 11:40 am (CST)

Title: Weak Stage Order Barriers for Runge-Kutta Schemes

Abstract: Runge-Kutta (RK) methods may exhibit order reduction when they are applied to solve certain stiff problems. Methods that satisfy certain weak stage order (WSO) conditions provide an avenue to avoid order reduction for linear problems with time-independent operators. Unlike high stage order, weak stage order can be achieved with diagonally implicit RK (DIRK) methods.

This talk will present the first general order barrier bounds relating the WSO of a scheme to its classical order and number of stages. These bounds characterize the fundamental accuracy limit of RK methods applied to stiff linear problems. New necessary conditions are also established on how one needs to split the spectrum of the Butcher matrix A so as to devise schemes with high WSO – which we use to construct new families of high WSO schemes. The key mathematical ideas are to recast WSO into a pair of orthogonal invarariant subspaces, and perform calculations modulo minimal polynomials. We also provide new formulas for the RK stability function in terms of a family of polynomials which are “orthogonal” with respect to a linear functional whose moment generating functions is related to the exponential function.

A portion of this work was performed at the Center for Nonlinear Analysis at Carnegie Mellon University. This work is in collaboration with A. Biswas, D. Ketcheson, B. Seibold.

Steven Roberts, Lawrence Livermore National Laboratory

Time: 11:40 am - 12:05 pm (CST)

TitleA Parallel Ensemble Approach to Constructing Stable, High-Order Time-Steppers

Abstract: With linear multistep, extrapolation, and deferred correction methods, it is possible to systematically generate time integration schemes of arbitrarily high order. Unfortunately, the linear stability can be difficult to analyze and often degrades as the order increases. For example, the stability regions of implicit methods tend to “pinch in” on the imaginary axis as their order increases.This talk will introduce parallel ensemble general linear methods:a family of integrators with stability regions that are invariant with respect to the order of accuracy. As the name suggests, the stages of these methods can be evaluated in parallel. Parallel ensemble methods also naturally extend to implicit-explicit, multirate, and alternating direction implicit methods while maintaining their excellent stability properties. We will conclude with numerical experiments to examine the convergence and efficiency properties.