Crank-Nicholson discretization scheme

Equation (8.25) represents a set of partial differential equations (PDEs) that must be solved in one space dimension and propagated in time. Typically, one discretizes the space variable 0 < x < 1 into n intervals. If the equations are linear, one can write difference equations at each point and solve the resulting matrix equation. The solution is progressed in time by also discretizing the time derivative with a Crank-Nicholson-like scheme. If the equations are nonlinear, the situation is more complicated. The appendix of Electrochemical Systems (Newman and Thomas-Alyea [25]) gives a discussion of numerical solution of partial differential equations. 

Application of the Crank-Nicholson method based on the spatial difference scheme (5.39) results in the following discretized form of the diffusion equation ... 

Fine resolution in the normal direction is necessary around the shear layers, and it gives severe limitation on the time step for numerical stability. Thus, it is preferred to compute the derivatives in the normal direction implicitly, while the derivatives in the streamwise direction are treated explicitly. This leads to a hybrid time-integration scheme with a low-storage third-order RK (RK3) scheme for explicitly treated terms and a second-order Crank-Nicholson scheme for implicitly treated terms. The overall accuracy is thus second order in time. The discretized Navier-Stokes equations have the forms ... 

Analogous stability analyses can be executed for the other time-discretization schemes as well. It is important to note here that although the von Neumann stabihty analysis yields a limiting time-step estimate to keep the round-off errors bounded, it does not preclude the occurrence of an bounded but unphysical solutions. A classical example is the Crank-Nicholson scheme, which from the von Neumann viewpoint is unconditionally stable, but can give rise to bounded unphysical solutions in case all the coefficients... 

The fractional step methods have become quite popular. To predict an accurate time history of the flow, higher order discretizations must be employed. Kim and Moin [106], for example, used a second order explicit Adams-Bashforth scheme for the convective terms and a second order implicit Crank-Nicholson scheme for the viscous terms. Boundary conditions for the intermediate velocity fields in timesplitting methods are generally a complex issue [3, 106]. There are many variations of the fractional step methods, due to a vast choice of approaches to time and space discretizations, but they are generally based on the principles described above. 

The governing equations are discretized by using the finite difference method. The Reynolds equation solution leads to solving a tri-diagonal system of linear equations. Using the semi-implicit scheme of Crank-Nicholson solves the energy equations in the film and in the rings. 

---