Explicit differencing methods

It can be noted that at least the explicit upwind method for the constant equation model gives the exact answer for CFL = 1, whereas the implicit upwind differencing method never does. The numerical viscosity of the implicit... 

Solution of Mathematical Model for Case 1. For the Case 1 solution iterative techniques were ruled unacceptable owing to the excessive time requirements of such methods. Several investigators (27, 28, 29, 30) working with similar noncoupled systems found that the Crank-Nicholson 6-point implicit differencing method (31) provided an excellent solution. For the solution of Equation (8) we decided to apply the Crank-Nicholson method to the second-order partials and corresponding explicit methods to the first-order partials. Nonlinear coefficients were treated in a special manner outlined by Reneau et al (5). 

For the time derivative there are several possible approximations. Three of these have been discussed in detail in Chapter 10 and are known as flie explicit forward differencing (FD) method, flie imphcit backwards differencing (BD) method and the trapezoidal rule (TP) which averages flie time derivative between two successive time points. From the discussion of fliese methods in Section 10.1, one would expect different long term stability results for each of fliese methods and this is certainly the case for partial differential equations as well as single variable differential equations. The forward and backwards time differencing methods leads to the set of equations ... 

This equation provides an appropriate replacement for a second-order time derivative with an explicit expression of the function value at the final time point and a function (17 ) fliat depends on the function value, first derivative and second derivative at the initial time point all of which are assumed to be known from a valid solution previously obtained. This is again identical to the second order time differencing method discussed in Section 10.7 with respect to single variable differential equations. 

Implicit methods are a bit more complicated to implement, but they are highly stable compared to explicit methods. For a linear system of equations, such as the present problem, there is no stability restriction at all. That is, the method will produce stable solutions for any value of the time step, including dt - oo. For nonlinear problems, or for higher-order time differencing, there is a stability limit. However, the implicit methods are always much more stable than their explicit counterparts. 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

Problem Consider laminar flow of a fluid over a flat plate. Use the explicit method of finite differencing to compute the x-component velocity profile within the boundary layer. 

The method of lines is one of a myriad possible schemes for solving PDEs. It has the merits of being/w/fy explicit and easy to implement. Its major weakness is a stability requirement that forces an overly small axial step size, Az. Some implicit differencing schemes avoid this limitation at the cost of solving sometimes large sets of linear algebraic equations. Chapter 16 describes the method of false transients as applied to PDEs that also allows more freedom in choosing Az. 

To facilitate the development of explicit and implicit methods, it is necessary to briefly consider the origins of interpolation and quadrature formulas (i.e., numerical approximation to integration). There are essentially two methods for performing the differencing operation (as a means to approximate differentiation) one is the forward difference, and the other is the backward difference. Only the backward difference is of use in the development of eiqjlicit and implicit methods. 

We solve the partial differential equations using a finite difference method that employs the usual staggered grid arrangement. AU convective terms are modelled using upwind differencing for stability. As far as possible we use explicit methods, treating source terms implicitly where appropriate, to ensure that positive quantities remain positive. ... 

As an example of a numerical differencing procedure in which derivatives are not explicitly calculated, the method of Ninomiya and Ferry requires values of G, spaced at equal intervals on a logarithmic frequency sc le above and below the frequency co = 1 /t corresponding to the value of t for which H is desired viz., at aw, and a w, with a suitable choice of a. Then... 

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