# The Nature of Mathematical Modelling

## 8.1

### (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.

See the Pen 8.1 by Ann (@beansbeansbeans) on CodePen.

## 8.2

### (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?

See the Pen 8.2a by Ann (@beansbeansbeans) on CodePen.

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

See the Pen 8.2b by Ann (@beansbeansbeans) on CodePen.

## 8.3

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

See the Pen 8.3 by Ann (@beansbeansbeans) on CodePen.

## 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.

See the Pen evEBNM by Ann (@beansbeansbeans) on CodePen.