Analytical-Numerical Solutions for the One Dimensional PBL

Analytical and propagator numerical methods are elaborated for

solution of Weng-Taylor turbulence model. In the Weng-Taylor

model the eddy viscosity coefficient is defined from additional

phenomenological consideration, which constitutes a turbulence

closure, and nonlinearly depend on velocities. In such type models

sharp vertical boundary layers causes difficulties for traditional

numerical methods. In this work a new numerical method is

proposed, based on analytical representation of Weng-Taylor model

solutions. It is shown that these analytical solutions of

constituted initial boundary value problem can be resolved by

additional solutions of system of ordinary differential equations.

This system of equations is solved numerically, by using

exponential/polinomial type substitution for turbulent fluxes. The

obtained numerical solution, precision and effectiveness are

compared to solution by using our numerical propagator method.

The propagator method exploits an approach of a non-standard

representation of the time derivative, by applying the derivative

to the solution given as the product of two functions, where one

of them is a propagator function. Propagator function is chosen in

nonlocal way, and with respect to the solution, a new finite

volume difference scheme is obtained. Stability of the scheme is

investigated, by using analytical solutions/estimations of

Weng-Taylor model. It is shown, that stability restrictions for

the propagator scheme becomes more weaker comparison to

traditional semi-implicit difference schemes. There are some

regions of the model coefficients, for which elaborated propagator

difference scheme becomes absolutely stable. It is proven that the

scheme is unconditionally monotonic. The scheme has the first

order in time and the second order truncation errors in space.