![]() ![]() Consider as an example the Poisson equation in three dimensions: This is especially true when solving multi-dimensional problems. However, for very large systems, matrix inversion becomes an expensive operation in terms of computational time and memory. ![]() When the size of the matrix is not too large, one can rely on efficient direct solvers. Their implementation is a bit more complicated in the sense that they require the inversion of a matrix. In the previous chapter we have discussed how to discretize two examples of partial differential equations: the one dimensional first order wave equation and the heat equation.įor the heat equation, the stability criteria requires a strong restriction on the time step and implicit methods offer a significant reduction in computational cost compared to explicit methods. ![]() insert ( 0, './modules' ) # Function to compute an error in L2 norm from norms import l2_diff % matplotlib inline Import numpy as np import matplotlib.pyplot as plt from scipy.sparse import diags import sys sys. ![]()
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