Application of Finite Difference Method to Solving Two-Point Boundary Value Problems in Spatial Dimensions
The finite difference method is a common numerical solution method for solving mathematical problems such as partial differential equations. In the spatial dimension, the finite difference method can be used to solve two-point boundary value problems with known boundary conditions. This article introduces the fundamentals of the finite difference method and provides corresponding source code examples.
The basic principle of the finite difference method is to discretize the solution area into a finite number of grid points, and then use the difference approximation to replace the derivatives in the partial differential equations to obtain a set of discrete algebraic equations. For 2D problems, we can discretize using a 2D grid, where each grid point represents a location in the solution domain.
Consider a simple two-dimensional problem, such as solving Laplace's equation:
∇²u = 0
Among them, u is the function to be solved, and ∇² represents the Laplacian operator. Suppose we solve this equation in a rectangular area Ω, the two points on the boundary are A and B respectively, and the known boundary conditions are:
u(A) = u_A, u(B) = u_B
In order to solve this problem using the finite difference method, we first need to discretize the solution domain Ω into a two-dimensional grid. Assuming that the width of Ω is Lx and the height is Ly, we can divide it into Nx grid points along the x-axis direction and Ny grid points along the y-axis direction.
Define the coordinates of each grid point as:
x_i = i * Δx, where i = 0, 1, 2, …, Nx, Δx = Lx / Nx
y_j = j * Δy, where j = 0, 1, 2, … , Ny, Δy = Ly / Ny
On the discretized grid, we can use u_{i,j} to represent the function value at the grid point (x_i, y_j), where i = 0, 1, 2, ..., Nx, j = 0, 1 , 2, …, Ny.
Difference Approximation According to Laplace's Equation