The CSE core is required for all CSE students in M.S. and Ph.D. programs. It consists of six one-quarter courses which are distributed over the three areas of numerical methods, parallel computing and applied mathematics.

### Numerical Methods:

A broad-based introductory graduate-level numerical analysis course (CHE 230D/MEE 244D) is available for students who do not have the background to begin the core CSE graduate sequence in numerical methods immediately.

CSE students must take at least three of the following courses. The courses need not be taken in sequence, although it may be advantageous to do so. These courses are cross-listed with Math, CS, ECE and CHE.

#### CS 211/MATH 206/MEE 210/ECEĀ 210 Sequence: Numerical Methods in Computational Science and Engineering

*Prerequisite: Consent of instructor. Students should be proficient in basic numerical methods, linear algebra, mathematically rigorous proofs, and some programming language.*

**210A Matrix Analysis and Computation**

Graduate level-matrix theory with introduction to matrix computations. SVD's, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.

**210B Numerical Simulation**

Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.

**210C Numerical Solution of Partial Differential Equations Finite Difference Methods**

Finite difference methods for hyperbolic, parabolic and elliptic PDEs, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.

**210D Numerical Solution of Partial Differential Equations Finite Element Methods**

Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptic partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.

### Parallel Computing:

CSE students must take the following: CS 240A

**240A. High-Performance Parallel Systems and Languages***Prerequisites: Computer Science 154 and 160.*

Overview of parallel architectures and communication; parallel programming paradigms; performance models of parallel computation; parallel algorithms and applications; systems and compilers for parallel languages.

### Applied Mathematics:

CSE students whose home department is not Mathematics must take one of the following sequences:

#### CHE 230A-B / MEE 244A-B Advanced Theoretical Methods in Engineering

**CHE 230A. Advanced Theoretical Methods in Engineering***Prerequisite: consent of instructor.*

Same course as ME 244A.

Methods of solution of partial differential equations and boundary value problems. Linear vector and function spaces, generalized Fourier analysis, Sturm-Liouville theory, calculus of variations, and conformal mapping techniques.

**CHE 230B. Advanced Theoretical Methods in Engineering***Prerequisites: Chemical Engineering 230A and consent of instructor.*

Same course as ME 244B.

Advanced mathematical methods for engineers and scientists. Complex analysis, integral equations and Green's functions. Asymptotic analysis of integrals and sums. Boundary layer methods and WKB theory.

#### Math 214A-B Ordinary Differential Equations, Chaotic Dynamics and Bifurcation Theory

**214A. Ordinary Differential Equations***Prerequisite: Not open to mathematics majors.*

Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.

**214B. Chaotic Dynamics and Bifurcation Theory***Prerequisite: Not open to mathematics majors.*

Hyperbolic structure and chaos; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.

#### Math 215A-B Partial Differential Equations, Fourier Transform and Numerical Methods

**215A. Partial Differential Equations***Prerequisite: Not open to mathematics majors.*

Wave, heat, and potential equations.

**215B. Fourier Series and Numerical Methods***Prerequisite: Not open to mathematics majors.*

Fourier series; generalized functions; and numerical methods.

Advanced courses may be substituted, with approval, as follows: Math 243 instead of Math 214, and Math 246 instead of Math 215.

### Students whose home department is Mathematics must take a two course sequence from either Mathematics 243A-B or Mathematics 246A-B.

**243A. Ordinary Differential Equations***Prerequisite: Mathematics 118A-B-C.*

Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant mainfolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.

**243B. Ordinary Differential Equations***Prerequisite: Mathematics 118A-B-C.*

Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant mainfolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.

**246A. Partial Differential Equations***Prerequisite: Mathematics 201A-B-C.*

First-order nonlinear equations; the Cauchy problem, elements of distribution theory and Sobolev spaces; the heat, wave, and Laplace equations; addition topics such as quasilinear hyperbolic systems, elliptic regularity theory.

**246B. Partial Differential Equations***Prerequisite: Mathematics 201A-B-C.*

First-order nonlinear equations; the Cauchy problem, elements of distribution theory and Sobolev spaces; the heat, wave, and Laplace equations; addition topics such as quasilinear hyperbolic systems, elliptic regularity theory.