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Computational Physics Lectures: Partial differential equations
Contents
Famous PDEs
Famous PDEs, two dimension
Famous PDEs, diffusion equation
Famous PDEs, Laplace's equation
Famous PDEs, Helmholtz' equation
Famous PDEs, Schroedinger's equation in two dimensions
Famous PDEs, Maxwell's equations
Famous PDEs, Euler's equations
Famous PDEs, the Navier-Stokes' equations
Famous PDEs, general equation in two dimensions
Diffusion equation
Diffusion equation
Diffusion equation, famous laws
Diffusion equation, heat equation
Diffusion equation, heat equation in one dimension
Diffusion equation, dimensionless form
Explicit Scheme
Explicit Scheme, boundary conditions
Explicit Scheme, algorithm
Explicit Scheme, simplifications
Explicit Scheme, solving the equations
Explicit Scheme, simple case
Explicit Scheme, matrix-vector formulation
Explicit Scheme, sketch of code
Explicit Scheme, stability condition
Explicit Scheme, spectral radius and stability
Explicit Scheme, eigenvalues and stability
Explicit Scheme, final stability analysis
Explicit Scheme, stability condition
Explicit Scheme, general tridiagonal matrix
Implicit Scheme
Implicit Scheme
Implicit Scheme
Implicit Scheme
Implicit Scheme
Program Example for Implicit Equation
Crank-Nicolson scheme
Derivation of CN scheme
Taylor expansions
Error in CN scheme
Truncation errors and stability
Rewrite of CN scheme
Final CN equations
Parts of Code for the Crank-Nicolson Scheme
Parts of Code for the Crank-Nicolson Scheme
Python code for solving the one-dimensional diffusion equation
Solution for the One-dimensional Diffusion Equation
Solution for the One-dimensional Diffusion Equation
Solution for the One-dimensional Diffusion Equation
Solution for the One-dimensional Diffusion Equation
Solution for the One-dimensional Diffusion Equation
Explict scheme for the diffusion equation in two dimensions
Explict scheme for the diffusion equation in two dimensions
Laplace's and Poisson's Equations
Laplace's and Poisson's Equations, discretized version
Laplace's and Poisson's Equations, final discretized version
Laplace's and Poisson's Equations, boundary conditions
Scheme for solving Laplace's (Poisson's) equation
Scheme for solving Laplace's (Poisson's) equation
Scheme for solving Laplace's (Poisson's) equation
Scheme for solving Laplace's (Poisson's) equation
Scheme for solving Laplace's (Poisson's) equation using Jacobi's iterative method
Scheme for solving Laplace's (Poisson's) equation, with Jacobi's method
Scheme for solving Laplace's (Poisson's) equation, final rewrite
Jacobi Algorithm for solving Laplace's Equation
Jacobi Algorithm for solving Laplace's Equation
Jacobi Algorithm for solving Laplace's Equation, the algorithm
Jacobi Algorithm for solving Laplace's Equation, simple example
Jacobi Algorithm for solving Laplace's Equation, to observe
Python code for solving the two-dimensional Laplace equation
Jacobi's algorithm extended to the diffusion equation in two dimensions
Jacobi's algorithm extended to the diffusion equation in two dimensions
Jacobi's algorithm extended to the diffusion equation in two dimensions, discretizing
Jacobi's algorithm extended to the diffusion equation in two dimensions, the second derivative
Jacobi's algorithm extended to the diffusion equation in two dimensions
"Solving project 1 again but now with Jacobi's method":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/Jacobi.cpp"
"Program to solve Jacobi's method in two dimension":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/diffusion2dim.cpp"
"The Jacobi solver function":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/diffusion2dim.cpp"
"Parallel Jacobi":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/MPIdiffusion.cpp"
"Parallel Jacobi":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/MPIdiffusion.cpp"
"Parallel Jacobi":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/LecturePrograms/programs/PDE/cpp/OpenMPdiffusion.cpp"
Wave Equation in two Dimensions
Wave Equation in two Dimensions, discretizing
Wave Equation in two Dimensions
Wave Equation in two Dimensions
Wave Equation in two Dimensions
Wave Equation in two Dimensions
Wave Equation in two Dimensions
Analytical Solution for the two-dimensional wave equation
Analytical Solution for the two-dimensional wave equation, first step
Analytical Solution for the two-dimensional wave equation,
Analytical Solution for the two-dimensional wave equation, separation of variables
Analytical Solution for the two-dimensional wave equation, separation of variables
Analytical Solution for the two-dimensional wave equation, boundary conditions
Analytical Solution for the two-dimensional wave equation, separation of variables and solutions
Analytical Solution for the two-dimensional wave equation, final steps
Python code for solving the two-dimensional wave equation
Famous PDEs, Maxwell's equations
Important systems of linear partial differential equations are the famous Maxwell equations
∂
E
∂
t
=
c
u
r
l
B
,
and
-\mathrm{curl} \mathbf{E} = \mathbf{B}
and
\mathrm{div} \mathbf{E} = \mathrm{div}\mathbf{B} = 0.
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