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This course addresses graduate students of all fields who are interested in numerical methods for partial differential equations. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. It particularly focus on a qualitative understanding of the considered partial differential equation, fundamentals of finite difference, finite volume, finite element, and spectral methods, and important concepts such as stability.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer need to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: MIT

Course Curriculum

Module: 01
Fundamental concepts and examples 00:05:00
Well-posedness and Fourier methods for linear initial value problems 00:05:00
Laplace and Poisson equation 00:05:00
Heat equation, transport equation, wave equation 00:05:00
General finite difference approach and Poisson equation 00:10:00
Elliptic equations and errors, stability, Lax equivalence theorem 00:05:00
Module: 02
Spectral methods 00:10:00
Spectral methods Part 2 00:10:00
Elliptic equations and linear systems 00:05:00
Efficient methods for sparse linear systems Multigrid 00:10:00
Efficient methods for sparse linear systems Krylov methods 00:05:00
Ordinary differential equations 00:05:00
Module: 03
Stability for ODE and von Neumann stability analysis 00:05:00
Advection equation and modified equation 00:05:00
Advection equation and ENO/WENO 00:10:00
Conservation laws Theory 00:05:00
Conservation laws Numerical methods 00:05:00
Conservation laws High resolution methods 00:10:00
Module: 04
Operator splitting, fractional steps 00:05:00
Systems of IVP, wave equation, leapfrog, staggered grids 00:05:00
Level set method 00:05:00
Navier-Stokes equation Finite difference methods 00:05:00
Navier-Stokes equation Pseudospectral methods 00:05:00
Particle methods 00:10:00
Assessment
Submit Your Assignment 00:00:00
Certification 00:00:00

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