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Conservative Linear Multistep Methods For Time-dependent PDEs

Project

Project Details

Program
Applied Mathematics and Computer Science
Field of Study
Applied mathematics
Division
Computer, Electrical and Mathematical Sciences and Engineering
Faculty Lab Link
http://numerics.kaust.edu.sa

Project Description

Many important physical phenomena are modeled by partial differential equations (PDEs) and possess physically significant conserved quantities, such as momentum or energy. Numerical methods do not typically preserve these conserved quantities exactly, and as a result numerical solutions may deviate both qualitatively and quantitatively from the true solutions of the model. In this project we will develop a class of methods that enforce exact conservation (up to rounding errors) of a desired set of quantities. Conservation will be achieved through the technique known as multiple relaxation, which has already been developed in the context of Runge-Kutta methods. Here we will apply similar ideas to the class of linear multistep methods, thus obtaining conservative high-order methods that require only a single derivative evaluation per step.

About the Researcher

David Ketcheson
Professor, Applied Mathematics and Computational Science
Computer, Electrical and Mathematical Science and Engineering Division

Affiliations

Education Profile

  • PhD University of Washington, USA, 2009
  • MS University of Washington, USA, 2008
  • BS Brigham Young University, USA, 2004

Research Interests

a€‹Professor Ketcheson's research interests are in the areas of numerical analysis and hyperbolic PDEs. His work includes development of efficient time integration methods, wave propagation algorithms, and modeling of wave phenomena in heterogeneous media.

Selected Publications

  • D.I. Ketcheson. ""Runge-Kutta methods with minimum storage implementations"". Journal of Computational Physics 229(5):1763 - 1773, 2010
  • S. Gottlieb, D.I. Ketcheson, and C.W. Shu. ""High order strong stability preserving time discretizations"". Journal of Scientific Computing, 38(3):251-289, 2009.
  • D.I. Ketcheson. C.B. Macdonald, and S. Gottlieb. ""Optimal implicit strong stability preserving Runge-Kutta methods"". Applied Numerical Mathematics, 59(2):373-392, 2009.
  • D.I. Ketcheson. ""Highly efficient strong stability preserving Runge-Kutta methods with low-storage implementations"". SIAM Journal on Scientific Computing, 30(4):2113-2136, doi: 10.1137/07070485X, 2008.
  • D.I. Ketcheson. ""Computation of optimal monotonicity preserving general linear methods"". Mathematics of Computation 78(267):1497-1513, 2009.

Desired Project Deliverables

The student will develop a Python code that implements multiple relaxation and is capable of solving a range of ODEs and PDEs.

Additionally, the student will take the lead in preparing a paper for publication.

Recommended Student Background

Differential equations
Programming
Numerical Methods
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3-6 months
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Cumulative GPA
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