The Nature of Mathematical Modelling

8.1

Consider the 1D wave equation

(f) Numerically solve the wave equation for the evolution from an initial condition with u = 0 except for one nonzero node, and verify the stability criterion.

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8.2

Write a program to solve a 1D diffusion problem on a lattice of 500 sites, with an initial condition of zero at all the sites, except the central site which starts at the value 1.0. Take D = ∆x = 1, and use fixed boundary conditions set equal to zero.

(a) Use the explicit finite difference scheme, and look at the behavior for ∆t = 1, 0.5, and 0.1. What step size is required by the Courant condition?

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(b) Now repeat this using implicit finite differences and compare the stability.

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8.3

Use ADI to solve a 2D diffusion problem on a lattice, starting with randomly seeded values.

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8.4

Use SOR to solve Laplace’s equation in 2D, with boundary conditions uj,1 = u1,k = 0, uN,k = −1, uj,N = 1, and explore how the convergence rate depends on α, and how the best choice for α depends on the lattice size.

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