## 1 Introduction  
The phase-locked loop or phase lock loop (PLL) [1; 2; 3; 4] is basically a nonlinear feedback control system that tracks the phase of the input signal by detecting the phase error and then adjusting the phase of the output. There are several different types; the simplest is an electronic circuit consisting of a variable frequency oscillator and a phase detector in a feedback loop. The oscillator generates a periodic signal, and the phase detector compares the phase of that signal with the phase of the input periodic signal, adjusting the oscillator to keep the phases matched. The GPS receiver [5; 6; 7; 8; 9] contains a code tracking loop and a carrier tracking loop for tracking the Doppler-shifted carrier simultaneously. A carrier-tracking loop tracks the carrier phase and the code-tracking loop tracks the signal code to within a small fraction of the chip duration. Most GPS receiver designs have a mode of operation that employs a non-coherent delay-locked loop (DLL) for code tracking and a Costas PLL for tracking the Doppler-shifted carrier (the frequency change due to vehicle dynamics). The pseudoranges obtained from the code tracking provide a position fix, while the pseudorange rate (or delta range) estimates obtained from the Costas loop provide a velocity fix, which is accomplished by counting the number of Doppler frequency shifts of carrier cycles that occur over a finite time interval. The statistics of a first-order PLL output with a white noise input have been shown to be a first-order Markov process. Furthermore, the time varying probability density for any Markov process can be described by the Fokker-Planck (FP) equation [10; 11; 12; 13; 14]. The FP equation is a second-order nonlinear, partial differential equation (PDE) that describes the time propagation of probability density functions and the transition probability for these stochastic processes. [PERSON] employed the FP equation to describe the variation of probability distribution of the phase error over time. He attempted to solve that equation for time approaching infinity to obtain the steady-state phase-error probability distribution [1; 10]. Since the analytical solution to the FP equation is relatively difficult, the numerical approach provides a good alternative. In fact, the transport (advection-diffusion) equation that frequently appears in the field of fluid dynamics [15; 16; 17; 18] bears a very good analogy with the modified FP equation. Therefore, the transport PDE is selected as the model equation for the modified FP equation. Among the available numerical methods [19; 20], the finite difference technique is the most attractive one to solve this type of equation. Explicit and implicit methods are approaches used in the numerical analysis for obtaining numerical approximations to the solutions of differential equations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one. The problem of choosing appropriate discretization is present in all finite difference methods--both explicit and implicit. The requirement based on the explicit methods places a serve restriction on the grid spacing size in the time direction with a resultant increase in computation. On the other hand, implicit finite-difference methods are unconditionally stable and need not satisfy this constrain. Therefore, the implicit finite difference methods are computationally advantageous methods with respect to the explicit ones for solving the PDE's. Implicit methods employed in the work include the backward-time/central space (BTCS) implicit method and the Crank-Nicolson (CN) method, which is second-order accurate both in time and space domains. The CN method offers a considerable improvement over the FTCS and BTCS implicit schemes, which are only first-order accurate in time.