Method of lines

Appendix 8.3 describes how finite differences can be used to approximate derivatives. Divide the radius of the tube into / increments, Ar = Rfl. For the radial direction we use second-order, central differences  

The PDF has been converted to a set of ODEs. Each version of Equation 8.30 governs the behavior of a as it evolves along a line of constant r. The independent variable is z. The main dependent variable is a(r) at location r. Equation 8.30 contains other dependent variables, a r + Ar) and a r — Ar), which are the main dependent variables along their own Tines. These side variables couple the set of equations so that they must be solved simultaneously. 

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. 

When using finite differences to solve the unsteady heat conduction problem, another approach involves writing finite difference equations at each grid point (node) only for the spatial variables while leaving the time derivative intact. This leads, generally, to a large number of simultaneous ODEs, which can be solved by, for example, a Runge-Kutta method. However, one must be careful since this set of ODEs can be stiff. Consider the same one-dimensional, unsteady state heat conduction problem as solved in Examples 8.1 and 8.2. This problem is solved by the method of lines in the next example. 

Applying a second-order correct finite difference analog for the spatial derivative term at line / gives 

At the left boundary, u,, = 0 because of the left boundary condition. So, the ODE when / = 1 becomes 

And the last equation (/ = n - 1) is as follows because the right-hand boundary condition is 1  

Shown below is a VBA subprogram FCalc that would be called by an ODE solver such as a Runge-Kutta method. The subprogram implements the right-hand-side functions of Equations 8.20 through 8.22. 

---

Schiesser, W. E. The Numerical Method of Lines. Academic Press (1991). 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

The Grank-Nicholson implicit method and the method of lines for numerical solution of these equations do not restrict the racial and axial increments as Eq. (P) does. They are more involved procedures, but the burden is placed on the computer in all cases. 

Example 8.5 Use the method of lines combined with Euler s method to determine the mixing-cup average outlet for the reactor of Example 8.4. ... 

The zero slope boundary condition at = 0 assumes S5anmetry with respect to the centerline. The mathematics are then entirely analogous to those for the tubular geometries considered previously. Applying the method of lines gives... 

The method of lines formulation for solving Equation (8.52) does not require that T aii be constant, but allows T aiiiz) to be an arbitrary function of axial position. A new value of T aii may be used at each step in the calculations, just as a new may be assigned at each step (subject to the stability criterion). The design engineer is thus free to pick a T au z) that optimizes reactor performance. 

The boundary conditions are unchanged. The method of lines solution continues to use a second-order approximation for dajdr and merely adds a Vr term to the coefficients for the points at r Ar. 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz. ... 

The overall solution is based on the method of lines discussed in Chapter 8. The resulting DDEs can then be solved by any convenient method. Appendix 13.2 gives an Excel macro that solves the DDEs using Euler s method. Figure 13.9 shows the behavior of the streamlines. 

A 5-point finite difference scheme along with method of lines was used to transform the partial differential Equations 4-6 into a system of first-order differential and algebraic equations. The final form of the governing equations is given below with the terms defined in the notation section. 

Schiesser, W.E. (1991) The Numerical Method of Lines Integration of Partial... 

For both the finite difference and weighted residual methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear s method are popular. 

In this book PDEs appear primarily in Section 8.1 and problem section P8.01. Some simpler methods of solution are mentioned there Separation of variables, application of finite differences and method of lines. Analytical solutions can be made of some idealized cases, usually in terms of infinite series, but the main emphasis in this area is on numerical procedures. Beyond the brief statements in Chapter 8, this material is outside the range of this book. Further examples are treated by WALAS (Modeling with Differential Equations in Chemical Engineering, 1991). 

Apply the method of lines to the solution of the unsteady state dispersion reaction equation with closed end boundary conditions for which the partial differential equation for a second order reaction is,... 

Formulate the solution by the method of lines for a steady state reaction in a vessel where dispersion occurs radially and axially. 

Analytical solution is possible only for first or zero order. Otherwise a numerical solution by finite differences, method of lines or finite elements is required. The analytical solution proceeds by the method of separation of variables which converts the PDE into one ODE with variables separable and the other a Bessel equation. The final solution is an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

The method of lines reduces a partial differential equation to a system of ordinary differential equations which can be solved by readily available software. It is applicable to PDEs that have only the first derivative of one of the variables, for example,... 

The method of lines replaces a partial differential equation with a set of ordinary differential equations. In an equation like that of P8.01.01, for instance,... 

Apply the method of lines to the heat and material balances of P8.01.04. The differential equations that apply except at the center and the wall are,... 

W. Pascher and R. Pregla, Vectorial analysis of bends in optical strip waveguides by the method of lines, Radio Science 28, 1229-1233 (1993). 

R. Pregla, The method of lines for the analysis of dielectric waveguide bends, J. Lightwave Technol. 14, 634-639 (1996). 

W. Pascher, Analysis of Waveguide Bends and Circuits by the Method of Lines and the Generalized Multipole Technique, Assoc. Prof, thesis, (FemUniversitat Hagen, Germany, 1998). 

R. Pregla, and W. Pascher, The method of lines, in T. Itoh (ed.). Numerical Technics for Microwave and Millimeter Wave Passive Structures, Wiley, New York, 381-446 (1989). 

R. Pregla, MoL-BPM Method of Lines Based Beam Propagation Method, in W. P. Huang (Ed.), Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices, Progress in Electromagnetic Research 11 (Elsevier, New York, 1995). 

---