MTH 656/659: Advanced Subjects in Numerical Analysis

Professor:

Dr. Vrushali Bokil
Office: Kidder 048
Phone: 737-2609
Email: Click to email: bokilv at math dot oregonstate dot edu
Office Hours: MF 1:00-1:50 pm, W: 11:00am-11:50am or by Appt.

Course Information:

Students should check this class website for general information, announcements, assignments, homework due dates, exam dates and other policies regarding the class.
Time/Classroom: MWF 2:00-2:50 pm BEXL 328
Registration Details: Math 656 (CRN:53181, Section:001), Math 659 (CRN:53961, Section:001)
Course Description:
The efficient numerical solution of transient wave equations used to model acoustic, elastic or electromagnetic wave propagation problems remains a challenge both for academic research and industry. In this course, we will survey a variety of techniques for the numerical solution of the basic models for linear wave propagation including high order finite difference methods, mass lumped (spectral) finite elements, as well as discontinuous Galerkin methods. Concepts important to numerical methods such as consistency, stability and convergence will be considered for the various numerical methods, and concepts specific to efficient numerical development of wave problems such as numerical dispersion error and anisotropy will be introduced. As time permits we will also consider additional topics including models for nonlinear wave propagation and their numerical treatment, multiscale numerical modeling of wave problems, among others. An important part of the course is implementation of finite difference and finite element methods using MATLAB. Thus, the assignments for this course will be partly theoretical and partly computational.

The intended audience for this course are graduate students of mathematics and other science and engineering disciplines. This is the third course in the advanced 654/5/6 sequence in numerical analysis, although courses may be taken in any order.

Learning Outcomes: After successfully completing this class students will:

• be able to derive and analyze standard finite difference methods for simple linear wave equations in 1 and 2 dimensions.
• Construct the variational or weak formuation for simple linear wave equations in 1 dimension in appropriate Hilbert spaces and derive fully discrete schemes using finite element approximations in space and finite differences in time.
• Understand the concepts of stability, consistency and convergence of numerical methods, the concepts of dissipation, and dispersion and how these relate to accuracy of finite difference and finite element methods.
• Be able to write simple codes in MATLAB to simulate finite difference and finite element methods for the PDEs considered, and understand basic computational aspects related to demonstrating accuracy, stability and convergence of numerical methods (CFL conditions, error rates and ratios)

Prerequisites: Familiarity with basic properties of differential equations (MTH 256) and matrices (MTH 341 or 306), some programming experience (preferably with MATLAB), and graduate standing. Familiarity with numerical methods (finite difference and finite element methods) for elliptic Partial differential equations and computation will be helpful; however we will develop the basics as necessary. Those who have taken the equivalent of our core numerical course (MTH 553) would be well-prepared. Students who are not sure about prerequisites are encouraged to talk to the instructor

Required Textbook and Reading Assignments: :
There is no required textbook. I will use material from several texts. Reading material will be assigned in class and posted on Canvas.

Day to day material covered will be posted on the Course Calendar. Questions not addressed during class time should be asked in office hours. Students are responsible for any material missed due to absence.

The course grade will be based on four written assignments this term, each worth 25% of the total grade. Written Assignments will be posted on Canvas and students can upload solutions on Canvas as well.