Study Notes

Course Projects Report

  • Numerical PDE Project1:This Report contains finite difference approximation to solve second-order ODE (BVP), verify the accuracy.

  • Numerical PDE Project2:This Report contains Jacobi Method, Gauss-Seidel Method to solve linear systems of equation which's corresponding to second-order ODE.

  • Numerical PDE Project3:This Report contains approximation of the solution to the 2nd-order ODE equation numerically using the Fast Fourier Transform (FFT) and verify 2nd-order accuracy.

  • Numerical PDE Project4:This Report contains Newton's method to solve the nonlinear system of equations from the nonlinear time-dependent PDEs.

  • Numerical PDE Project5:This Report contains Three schemes (Forward Time, Centered Space; Leapfrog; Lax-Friedrichs) to solve transport equation with periodic boundary condition; Von Neumann Analysis for FTCS; Find the order of accuracy of each method).

Take-home Notes

  • Homogenization hw1: Consider the propoties of weak convergence, as a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded. The norm is weakly lower-semicontinuous.

  • Homogenization hw2: Prove C_0^{\infty}(\Omega) is dense in L^p(\Omega);

  • Homogenization hw3: The energy minimization to deduce the Landau-Lifhiz equation;

  • Homogenization hw4: Asymptotics analysis on boundary layer;

  • Homogenization hw5: The multi-dimensional case for periodic composite materials;

  • Homogenization hw6: The Optimized problem of homogenization coefficients;

  • Homogenization hw7: The linear response representation.