A Numerical Method for solving the Periodic Burgers' Equation through a Stochastic Differential Equation

The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2021

Author dc:creator

• Shedlock, Andrew James

Chair dc:contributor.committeechair

• Borggaard, Jeffrey T.

Committee members dc:contributor.committeemember

• Rossi, John F.
• Zietsman, Lizette

• Stochastic Differential Equations
• Burgers equation
• Numerical Methods
• Finite Difference Method

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

vt_gsexam:30717

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/103947