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Document Type

Thesis - University Access Only

Award Date


Degree Name

Master of Science (MS)

Department / School

Mathematics and Statistics

First Advisor

Jung-Han Kimu


For certain formulations of partial differential equations, proper time-parallel pre conditioners can be successfully applied in space-time finite element simulations. Such an approach may enable the extraction of more parallelism to better utilize high performance computing resources. In this thesis, we have implemented a fully implicit numerical approach based on space-time finite element methods for the Klein-Gordon equation in the 1(space)+1(time) dimension. The proposed numerical method is applied to generate successful simulation results of spin-0 particle propagation in a charged scalar field. The time additive Schwarz method is vital to make successful simulations with KSP (Krylov Subspace Methods) solvers. Also, a fully implicit numerical approach based on the space-time finite element method is implemented for the semilinear wave equation in 1(space) + 1(time) dimensions to explore critical collapse and search for self-similar solutions. Previous work studied this behavior by exploring the threshold of singularity formation using time marching finite difference techniques while this work introduces an adaptive time parallel numerical method to the problem. The semilinear wave equation with a p = 7 term is examined in spherical symmetry. The impact of mesh refinement and the time additive Schwarz preconditioner in conjunction with Krylov Subspace Methods are examined. The time parallelizable algorithm is implemented through PETSc(Portable, Extensible, Toolkit for Scientific Computation, developed by Argonne National Laboratory).

Library of Congress Subject Headings

Wave equation
Klein-Gordon equation
Differential equations, Partial
Decomposition (Mathematics)


Includes bibliographical references (pages 82-85)



Number of Pages



South Dakota State University



Rights Statement

In Copyright