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Numerical Simulation Overview

Introduction

This folder contains learning materials for numerical simulation using Python. It covers the full range from basic ordinary differential equations (ODE) to magnetohydrodynamics (MHD) and plasma simulation.

Learning Roadmap

Basics (01-02)
    ↓
Ordinary Differential Equations ODE (03-06)
    ↓
Partial Differential Equations PDE Basics (07-08)
    ↓
Heat/Wave/Steady-State Equations (09-12)
    ↓
Computational Fluid Dynamics CFD (13-14)
    ↓
Electromagnetic Simulation (15-16)
    ↓
Magnetohydrodynamics MHD (17-18)
    ↓
Plasma Simulation (19)
    ↓
Monte Carlo Simulation (20)
    ↓
Spectral Methods (21)
    ↓
Finite Element Method (22)
    ↓
GPU Acceleration (23)
    ↓
Physics-Informed Neural Networks PINN (24)

File List

File Topic Key Content
01_Numerical_Analysis_Basics.md Numerical Analysis Basics Floating-point, error analysis, numerical differentiation/integration
02_Linear_Algebra_Review.md Linear Algebra Review Matrix operations, eigenvalues, decomposition (LU, QR, SVD)
03_ODE_Basics.md ODE Basics ODE concepts, initial value problem, analytical solutions
04_ODE_Numerical_Methods.md ODE Numerical Methods Euler, RK2, RK4, adaptive step
05_ODE_Advanced.md ODE Advanced Stiff problems, implicit methods, scipy.integrate
06_ODE_Systems.md Coupled ODE and Systems Lotka-Volterra, pendulum, chaotic systems (Lorenz)
07_PDE_Overview.md PDE Overview PDE classification, boundary conditions, initial conditions
08_Finite_Difference_Basics.md Finite Difference Basics Grid, discretization, stability conditions (CFL)
09_Heat_Equation.md Heat Equation 1D/2D heat conduction, explicit/implicit methods
10_Wave_Equation.md Wave Equation 1D/2D waves, boundary reflection, absorbing boundaries
11_Laplace_Poisson.md Laplace/Poisson Steady-state, iterative methods (Jacobi, Gauss-Seidel, SOR)
12_Advection_Equation.md Advection Equation Upwind, Lax-Wendroff, numerical dispersion/diffusion
13_CFD_Basics.md CFD Basics Fluid dynamics concepts, Navier-Stokes introduction
14_Incompressible_Flow.md Incompressible Flow Stream function-vorticity, pressure-velocity coupling, SIMPLE
15_Electromagnetics_Numerical.md Electromagnetics Numerical Maxwell equations, FDTD basics
16_FDTD_Implementation.md FDTD Implementation 1D/2D electromagnetic wave simulation, absorbing boundaries (PML)
17_MHD_Basics.md MHD Basic Theory Magnetohydrodynamics concepts, ideal MHD equations
18_MHD_Numerical_Methods.md MHD Numerical Methods Conservative form, Godunov method, MHD Riemann problem
19_Plasma_Simulation.md Plasma Simulation PIC method basics, particle-mesh interaction
20_Monte_Carlo_Simulation.md Monte Carlo Simulation Random number generation, MC integration, Ising model, option pricing, variance reduction
21_Spectral_Methods.md Spectral Methods Fourier spectral, FFT differentiation, Chebyshev collocation, dealiasing
22_Finite_Element_Method.md Finite Element Method Weak form, basis functions, stiffness matrix assembly, 1D/2D FEM
23_GPU_Acceleration.md GPU Acceleration CUDA basics, CuPy, GPU-accelerated PDE solvers, performance optimization
24_PINN.md Physics-Informed Neural Networks (PINN) Neural network solvers, loss functions from PDEs, automatic differentiation, inverse problems

Required Libraries

# Basic
pip install numpy scipy matplotlib

# Performance optimization (optional)
pip install numba

# 3D visualization (optional)
pip install mayavi

Library Roles

Library Purpose
NumPy Array operations, linear algebra
SciPy ODE solvers, sparse matrices, optimization
Matplotlib 2D visualization, animation
Numba JIT compilation, performance optimization

Recommended Learning Sequence

Stage 1: Basics (1-2 weeks)

  • 01_Numerical_Analysis_Basics.md
  • 02_Linear_Algebra_Review.md

Stage 2: ODE (2-3 weeks)

  • 03_ODE_Basics.md
  • 04_ODE_Numerical_Methods.md
  • 05_ODE_Advanced.md
  • 06_ODE_Systems.md

Stage 3: PDE Basics (2-3 weeks)

  • 07_PDE_Overview.md
  • 08_Finite_Difference_Basics.md
  • 09_Heat_Equation.md
  • 10_Wave_Equation.md

Stage 4: Steady-State and Advection (1-2 weeks)

  • 11_Laplace_Poisson.md
  • 12_Advection_Equation.md

Stage 5: CFD (2-3 weeks)

  • 13_CFD_Basics.md
  • 14_Incompressible_Flow.md

Stage 6: Electromagnetics (2 weeks)

  • 15_Electromagnetics_Numerical.md
  • 16_FDTD_Implementation.md

Stage 7: MHD and Plasma (3-4 weeks)

  • 17_MHD_Basics.md
  • 18_MHD_Numerical_Methods.md
  • 19_Plasma_Simulation.md

Stage 8: Stochastic Simulation (2 weeks)

  • 20_Monte_Carlo_Simulation.md

Stage 9: Advanced Methods (2-3 weeks)

  • 21_Spectral_Methods.md
  • 22_Finite_Element_Method.md

Stage 10: Modern Computational Techniques (2-3 weeks)

  • 23_GPU_Acceleration.md
  • 24_PINN.md

Prerequisites

  1. Python Basics: NumPy array operations
  2. Calculus: Differentiation, integration, partial derivatives
  3. Linear Algebra: Matrices, eigenvalues, decomposition
  4. Physics: Mechanics, basic electromagnetics (for CFD/MHD)

Simulation Code Structure Example

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

# 1. Parameter setup
nx, ny = 100, 100
dx, dy = 1.0, 1.0
dt = 0.01
n_steps = 1000

# 2. Initial conditions
u = np.zeros((nx, ny))

# 3. Time integration loop
for step in range(n_steps):
    # Apply boundary conditions
    # Calculate spatial derivatives
    # Time advancement
    pass

# 4. Result visualization
plt.imshow(u)
plt.colorbar()
plt.show()

References

Textbooks

  • Computational Physics - Mark Newman
  • Numerical Recipes - Press et al.
  • CFD Python (12 Steps to Navier-Stokes) - Lorena Barba

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