Math 6641
Advanced Numerical Methods for Partial Differential Equations
Course Information (Spring 2022)

Time:T R 15:30-16:45
Place:Skiles 268
Reference books:
J. W. Thomas, Introduction to Numerical Methods for Partial Differential Equations, Springer, ISBM 0-387-97999-9.
L. N. Trefethen, Spectral Methods in Matlab, SIAM.
S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Second Edition.
Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, 1992. ISBN 0521 345 146
R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, 1992.
Vidar Thomee, Galerkin finite element method for parabolic problems.
S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp., 1980.
C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni) , Lecture Notes in Mathematics, volume 1697, Springer, 1998, pp.325-432.
Instructor:Dr. Yingjie Liu
Office:Skiles 134
Office Hours: T R 16:45-17:45
Homework:Homeworks will be assigned once every couple of weeks and they must be turned in on time.
Grading:Homework 50%, final 50%.
Topics: Some recent development in high order methods for conservation laws and related equations will be discussed, such as FCT and flux limiter, MUSCL and slope limiter, finite difference and finite volume ENO/WENO schemes on rectangular and unstructured meshes; central scheme, central scheme on overlapping cells; discontinuous Galerkin method (DG), LDG and central DG on overlapping cells; Hierarchical reconstruction methods (HR) for reducing numerical artifacts (due to non smoothness of the solution) on unstructured triangular meshes; finite volume methods with HR etc. Some classical methods for solving Navier-Stokes equation of incompressible fluids will be introduced, such as the Marker-and-Cell method, projection method, mixed finite element methods etc. Interface tracking and capturing methods will be covered, including front-tracking, volume of fluids and the level set methods. Recent development on BFECC method and its limiting for solving time dependent Hamilton-Jacobi equations on regular or irregular meshes, and BFECC for solving Maxwell's equations for electromagnetic waves and for perfectly matched layers (PML). Recent development on neural network methods for solving conservation laws.
Homework 1, due Mar. 1, 2022.
Homework 2: due 3/31.