An Investigation Of Numerical Solution To Partial Differential Equations In Numerical Weather Prediction

Abstract

In this study, we begin by presenting an overview of the Numerical Weather Prediction process as used in the Unified Model of the Met Office in UK The primitive equations are the continuity equation, the momentum equations, equations for representation of moisture, the expression for the first law of thermodynamics and the equation of state. Discretisation of these equations is done using the two-time-level, off-centred, Semi-Implicit, Semi­ Lagrangian time discretisation scheme. This is preferred to the Eulerian decomposition in which advection terms abound, rendering it computationally inefficient. Consistency and stability of this scheme is analysed; the latter using the matrix method of stability analysis. Convergence of the SISL scheme is inferred by using Lax Equivalence Theorem.

The coupling of the discretised governing equations results in the Helmholtz equation whose solution yields the increment in pressure field, n'. To analyse the condition for stability of the Helmholtz equation we have used the Von Neumann approach, which shows that the spatial-steps chosen and the wave number are factors that affect stability. In the final analysis, the recurrence relation for a 2 - D Helmholtz equation is solved using Jacobi, Gauss-seidel, Successive-Over-Relaxation, Conjugate Gradient, Bi-Conjugate Gradient, Bi-Conjugate Gradient Stabilized, Quasi-Minimal Residual and Gradient Minimal Residual methods. Respective iteration time is also shown. We show that Bi-CGSTAB method is most efficient, followed by GMRES method. Finally, a visual aid for the solution of2-dimensional Helmholtz equation is shown.