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Integral equations, constructing O(N) unconditional stable methods for both linear and non-linear PDEs.

Andrew Christleib Department Chair, Department of Computational Mathematics, Science and Engineering; MSU Foundation Professor, Department of Mathematics

Michigan State University

Monday, March 22, 2021
3:30 p.m.–4:30 p.m.
Zoom Virtual Setting

Abstract:  In this talk we introduce a novel approach to constructing approximations to derivatives for both linear and non-linear PDEs based on integral operators.  These approximations lead to a class of unconditionally stable methods for linear PDEs and  methods that behaves unconditionally stable for non-linear PDEs.  The integral operators can be evaluated in time O(N) and do not require iteration for non-linear problems.  A simple way to think about the method is that the integral operators are directly related to the resolvent expansion of the differential operator.  We will demonstrate the method for a range of problems, including Maxwell’s equations,  Hamilton Jacobian problems and Nonlinear advection diffusion equations.


christliebBio:  Andrew Christlieb received his Ph.D. from the University of Wisconsin-Madison in 2001. Upon completing his Ph.D., he took a postdoc in the Aerospace Department at the University of Michigan with Iain Boyd, working on the simulation of micro air foils. He then transitioned to a postdoc in the Mathematics Department at the University of Michigan, where he worked with Robert Krasny on the development of mesh-free methods for plasma simulations. Since 2004, he has worked very closely with the RDHE group at the Air Force research labs on the development of new methods for particle simulations of plasmas. In 2006, Christlieb joined the mathematics department at Michigan State University. In 2006, he was awarded a summer faculty fellow from the Air Force to work with AFRL Edwards on modeling of electric pupation. In 2007, he received the Air Force Young Investigator Award for his work on the development of novel methods for simulating plasmas. From 2008-2012, Christlieb was an IPA for the directed energy group at Kirtland Air Force Base. In 2010, he was promoted to associate professor and in 2014 he was promoted to professor. In 2015, he was named an MSU Foundation Professor.

Christlieb has an active research group, focusing on multi-scale modeling, high order numerical methods and sub-linear lossy compression algorithms. He is currently advising 2 postdocs and 6 students. His former Ph.D. students have gone on to work at national labs, industry and in academia. He has been involved in the development of a host of high order Eulerian, Lagrangian and semi-Lagrangian conservative methods for the kinetic simulation of plasmas, as well as the development of high order finite difference constrained transport methods for the simulation of magnetohydrodynamics targeted at AMR codes and new implicit Maxwell solvers targeting scale separation in plasmas. Further, his group has done work on high order gradient stable methods for phase field models, including the 6th order functionalized Cahn Hilliard model.  Christlieb's group has been funded by AFOSR Computational Mathematics, AFOSR Physics and Electronics, AFRL RDHE, NSF Division of Mathematics and ORNL LDRD on scalable computing.