Course Name: Mathematics for Engineering (English)

Course Code: S131A035

Semester: 2

Credit: 4

Program: All

Course Module: Basic Science Courses

Responsible: Course Teaching Group

E-mail:

Department: School of Mathematics， Tianjin University

TimeAllocation(1 credit hour = 45 minutes)

Course Description

A graduated level mathematics course, which supports the engineering area.

Pre-requisite

Calculus, Differential Equations

Course Objectives

This course has two major topics:

1. Initial Value Problems

   1. Linear: Wave Equation, Heat Equation, Convection Equation

   2. Nonlinear: Conservation Laws, Navier-Stokes Equation

   3. Finite Difference Methods: Accuracy and Stability

   4. Lax Equivalence Theorem: CFL and Von Neumann Conditions

   5. Fourier Analysis: Diffusion, Dissipation, Dispersion

   6. Separation of Variables and Spectral Methods

2. Solution of Large Linear Systems

   1. Finite Differences, Finite Elements, Optimization

   2. Direct Methods: Reordering by Minimum Degree

   3. Iterative Methods and Preconditioning

      1. Simple Iteration (Jacobi, Gauss-Seidel, Incomplete LU)

      2. Krylov Methods: Arnoldi Orthogonalization

      3. Conjugate Gradients and GMRES

      4. Multigrid Methods

   4. Inverse Problems and Regularization

Course Syllabus

Section 3.5 - Finite Differences and Fast Poisson Solvers

Section 5.1- Finite Difference Methods

Section 5.2 - Accuracy and Stability for $u_t= cu_x$

Section 5.3 - The Wave Equation and Staggered Leapfrog

Section 5.4 - The Heat Equation and Convection-Diffusion

Section 5.5 - Difference Matrices and Eigenvalues

Section 5.6 - Nonlinear Flow and Conservation Laws

Section 5.7 - Level Sets and the Fast Marching Method

Section 6.1 - Elimination with Reordering

Section 6.2 - Iterative Methods

Section 6.3 - Multigrid Methods

Section 6.4 - Krylov Subspaces and Conjugate Gradients

Section 6.5 - The Saddle Point Stokes Problem

Section 7.1 - One Fundamental Example

Section 7.2 - Calculus of Variations

Textbooks & References

• Strang, Gilbert,Computational Science and Engineering. Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.

Grade Distribution

Homework, Projects and Final Exam

Capability Tasks

CT1: To understand basic science, and to have analytical ability and the ability to integrate related knowledge.

CT2: To apply relevant professional knowledge to the field of science and technology: understanding of the basic concepts and its connotation, application of different methods and concepts which have been learned, capability of judging the scope and limitations of such applications.

CT3: To grasp methodologies and engineering tools: identifying, utilizing and solving problems. Even if the students are not familiar with the content, they can turn to computer tools for systematic analysis.

Achievement

• To use Numerical methods for initial-value problems. -Level: A

• To be able to find solution of Large Linear Systems. -Level: A

• To solve engineering problem with mathematics modeling. -Level: A

Students: All programs, Year 1