Effects of time discretization on wave propagation

Abstract

For the standard methods of time discretization, discussed in sections 8.2 and 8.3, the numerical accuracy for linear wave propagation can be studied using the finite-difference analogue of (7.1):

$$\textup{v}^{n}=\rho ^{n}\;exp(ik_{1}x+ik_{2}y)\textup{v}_{1}$$

((9.1))

(on the right-hand side, n is an exponent). The eigenvector v ₁ of the matrix M ⁻¹ A in (8.1) is the same as in the semi-discretized continuous-time case. The error is therefore characterized by comparing ρⁿ to exp(-ivt) with t = n Δt and v is the wave frequency for the semi-discrete equations, as derived in chapter 7. Unless bottom friction plays a role (which is particularly so for flood waves), v is real so there is no wave damping in the semi-discrete case. If there is any wave damping, it is generated by time-differencing and you can define a damping factor as

$$d_{n}=|\rho |^{n}$$

((9.2))

The number of time steps is arbitrary in principle, but for purposes of comparison, it can be chosen to correspond to a physically meaningful period of time; here we choose one wave period of the semi-discrete solution:

$$n=\frac{2\pi}{v\Delta t}$$

The relative wave speed is defined as the ratio of the phase shifts during one time step:

$$c_{r}=\frac{arg(\rho)}{-v\Delta t}$$

((9.3))

With the expressions for ρ in (8.3),(8.9),(8.11) (forward Euler is not included as it is unstable for this case) and taking into account that the eigenvalues of M ⁻¹ A are μ = iv, the quantities d _n and c _r can be evaluated. Both should be as close to unity as possible for accuracy; the deviation from unity is plotted in fig. 9.1. You can use this for gravity, vorticity and Rossby waves by inserting the appropriate value for vΔt from chapter 7. Note that these errors come on top of those due to spatial discretization discussed there.