Numerical Weather Prediction

 Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. 

A more fundamental problem lies in the chaotic nature of the partial differential equations that govern the atmosphere. It is impossible to solve these equations exactly, and small errors grow with time (doubling about every five days). Present understanding is that this chaotic behavior limits accurate forecasts to about 14 days even with perfectly accurate input data and a flawless model. In addition, the partial differential equations used in the model need to be supplemented with parameterizations for solar radiation, moist processes (clouds and precipitation), heat exchange, soil, vegetation, surface water, and the effects of terrain.

There is only one method that ‘comes to mind’. It is called the semi-implicit method in the meteorology community. It is ‘clear’ that this can only be achieved by an unconditionally stable implicit time-stepping method (trapezoidal rule).

METHODS FOR SOLVING THE GRAVITY AND INERTIA-GRAVITY WAVE EQUATIONS

The semi-implicit method of Kwizak and Robert 

The splitting or Marchuk method

METHODS FOR SOLVING ELLIPTIC EQUATIONS

Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.It is a generalization of Laplace's equation, which is also frequently seen in physics

Richardsons Method

Liebmanns method

Southwell’s Residual Relaxation Method

Successive Over-Relaxation (SOR) Method

Fourier-Series Methods

Neumann Boundary conditions in SOR Method

Example of Relaxation Methods

SPECTRAL METHODS

Silbermann’s method (spherical harmonics)

THE ADVECTION EQUATION

Energy method

Advection is treated in a semi-Lagrangian fashion.