Solving Partial Differential Equations Using Finite Difference Method

We can employ the finite difference method to solve partial differential equations. This numerical approach discretizes the continuous domain into finite intervals, converting differential operators into difference equations. The implementation typically involves defining a computational grid and approximating derivatives using central, forward, or backward difference schemes. For the iterative solution, we apply the Gauss-Seidel method which starts with an initial guess and successively updates grid point values using neighboring computed values. In code implementation, this involves nested loops over grid points where each new value is immediately used in subsequent calculations within the same iteration. The convergence criterion is determined by calculating the infinity norm (maximum absolute row sum) of the matrix containing differences between solutions from two consecutive iterations. This can be implemented using a while loop that checks if the norm falls below a specified tolerance threshold. The algorithm structure generally follows: initialize grid, apply boundary conditions, iterate until convergence, with each iteration updating interior points using the discretized PDE formulation. This method is versatile for various PDE types (elliptic, parabolic, hyperbolic) and finds extensive applications in engineering simulations and scientific computing, particularly for problems with regular domains where structured grids are applicable. Proper implementation requires careful handling of boundary conditions and stability considerations based on the specific PDE characteristics.