numerics for geofluid dynamics

second-order linear and homogeneous PDE are from the kind:

\(.\\a \frac{\partial^2u}{\partial x^2}+b \frac{\partial^2 u}{\partial x \partial y}+c \frac{\partial ^2 u}{\partial y^2} + d \frac{\partial u}{\partial x}+ e \frac{\partial u}{\partial y}+ fu+g=0\)

resembles the equation for a conic section 

\(ax^2+bxy+cy^2+dx+ey+f=0\)

1. forward difference: \((\frac{du}{dx})_j\rightarrow \frac{u_{j+1}-u_j}{\Delta x}\)
2. centered difference: \((\frac{du}{dx})_j \rightarrow \frac{u_{j+1}-u_{j-1}}{2\Delta x}\)
3. backward difference: \((\frac{du}{dx})_j \rightarrow \frac{u_{j}-u_{j-1}}{\Delta x}\)

1. the error of the numerical solution is brought up by:
   1. inadequacy of the scheme 
   2. insufficient information about the initial consitions (only known at discrete time points) 

-> Thus, an increase of accuracy of the scheme improves only one of these two components, and the result is not too impressive. 

2.  for stability requirements, it is necessary to choose a time step significantly smaller than that required for adequate accuracy.

-> With timesteps usually chosen, other errors, for example in the space differencing, are much greater that those to the time differencing. 

NMiMaO, S.14

In mathematics, the Courant–Friedrichs–Lewy (CFL) condition is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically by the method of finite differences. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution.

If a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points.

\(c\leq \frac{\Delta x}{\Delta t}\)

consider the equation \(\frac{du}{dt}=f(u,t)\;\;where\;u=u(t)\)

if a value of f for integration is taken at time level n+1, than this scheme is implicit. 

if this is not the case, than the scheme is implicit. 

second order central:

\((\frac{d^2u}{dx^2 })_j \rightarrow \frac{u_{j+1}-2u_j+u_{j-1}}{\Delta x^2}\)

second order forward:

\((\frac{d^2u}{dx^2})_j \rightarrow \frac{u_{j+2}-2*u_{j+1}+ u_j}{\Delta x^2}\)

second order backward:

\((\frac{d^2u}{dx^2 })_j \rightarrow \frac{u_j-2*u_{j-1}+u_{j-2}}{\Delta x^2 }\)

\(\frac{\partial u}{\partial t}+c*\frac{\partial u}{\partial x}=0\)

\(\frac{u^{n+1}_j-u^n_j}{\Delta t}+c\frac{u^n_{j+1}-u^n_{j-1}}{2\Delta x}=0\)

A solution to a linear equation can be expressed as a Fourier series where each Fourier component is a solution. Thus, we can test the stability with one single Fourier component of the form:

\(.\\u^n_j=u_0*e^{ik(j\Delta x-C_Dn\Delta t)}\)

for the time derivatives: 

\(u^{n+m}{j}=u^n_j*\lambda ^m\)

----------------------------

\(u^{n+1}_j=\lambda u^n_j\;\;\;where\;\;\;\lambda=e^{-ikC_D\Delta t}\)

for stability it's necessary that:

\(|\lambda|=|e^{-ikC_D\Delta t}|\leq1\)

-----------------------

we get:

\(\lambda =1-i\frac{c\Delta t}{\Delta x}*sin(k\Delta x)\)

The absolute value of \(\lambda\) is:

\(|\lambda|^2=1+[\frac{c\Delta t}{\Delta x}*sin(k\Delta x)]^2 \)

\(|\lambda|>1\)

The scheme is unconditionally unstable!