Method of lines technique

The upper curve was calculated by the method of characteristics program and it exhibits a true limit cycle or sustained oscillation response (19). The middle curve was calculated by the distance method of lines program. The response is attenuated and stretched out. The final long term oscillations had random unequal periods and they were out of phase with the MOC results. The lower curve was calculated by the time method of lines program. The initial part of the response is similar to the DMOL results but then the temperature incorrectly levels out to a steady state condition. Thus, it was evident that the distance and time method of lines techniques were not as accurate as the method of characteristics procedure for calculating the gasifier step responses and they were discarded. 

Step change dynamic response runs revealed that the distance and time method of lines techniques were not as accurate as the method of characteristics procedure for calculating gasifier transients. Therefore, these two techniques were discarded and the remaining calculations were all done using the method of characteristics. 

In section 3.2.3, finite difference solutions were obtained for nonlinear boundary value problems. This is a straightforward and easy technique and can be used to obtain an initial guess for other sophisticated techniques. This technique is important because it forms the basis for the method of lines technique for solving linear and nonlinear partial differential equations (chapter 5 and 6). However, for stiff boundary value problems, this technique may not work and might demand prohibitively large number of node points. In addition, approximate initial guess should be provided for all the node points for stiff boundary value problems. 

Using the boundary conditions (equations (5.7) and (5.8)) the boundary values uo and Un+1 can be eliminated. Hence, the method of lines technique reduces the linear parabolic ODE partial differential equation (equation (5.1)) to a linear system of N coupled first order ordinary differential equations (equation (5.5)). Traditionally this linear system of ordinary differential equations is integrated numerically in time.[l] [2] [3] [4] However, since the governing equation (equation (5.5)) is linear, it can be written as a matrix differential equation (see section 2.1.2) ... 

The second approach, applied by Margolis (1978) and by Heimerl and Coffee (1980) to ozone decomposition flames, employs a method-of-lines technique. In combination with finite-element collocation methods this technique provides a general approach to the numerical solution of partial differential equations. Taking Eqs. (4.12) and (4.13) as the working examples... 

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. 

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). 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

The resulting set of model partial differential equations (PDEs) were solved numerically according to the method of lines, applying orthogonal collocation techniques to the discretization of the unknown variables along both the z and x coordinates and integrating the resulting ordinary differential equation (ODE) system in time. 

The method of lines is a computational technique that is particularly suited for solving coupled systems of parabolic partial-differential equations (PDE). The boundary-layer equations can be solved by the method of lines (MOL), although the task is facilitated considerably by casting the problem in a differential-algebraic setting [13]. As an introductory illustration, consider the heat equation... 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

This transformation allows for equal distribution in the y-space while concentrating the lines close to the x = L boundary. Parameter a sets the spacing of the lines. This technique is called MOL1D (Method Of Lines in 1 Dimension) and is suitable for solving parabolic and hyperbolic initial boundary value problems in one dimension. 

A common method for solving partial differential equations (PDEs) is known as the method of lines. Here, finite difference approximations for spatial derivatives are used to convert a PDE model to a large set of ordinary differential equations, which are then solved using any of the ODE integration techniques discussed earlier. 

In contrast to the method of line reversal described in the preceding section, the discussed technique can be applied even if the emission bands are very weak. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

In section 5.2.4, a stiff nonlinear PDE was solved using numerical method of lines. This stiff problem was handled by calling Maple s stiff solver. The temperature explodes after a certain time. The numerical method of lines (NMOL) technique was then extended to coupled nonlinear parabolic PDEs in section 5.2.5. By comparing with the analytical solution, we observed that NMOL predicts the behavior accurately. 

Both analytical and numerical methods of lines are presented in this chapter for elliptic partial differential equations. Semianalytical method, presented in this chapter is very powerful technique, and is valid for elliptic Partial differential equations with mixed boundaries also (Subramanian and White, 1999). Numerical method of lines presented in this chapter should be used with precaution, as it may not work for stiff problems. A total of seven examples were presented in this chapter. 

The three main formats for sample preparation used in drug-discovery are protein precipitation (PPT), SPE, and LLE. Several examples of off-line sample preparation have been reported and involve SPE [37,38,46,47], LLE [38,48], and PPT [39,49]. In each of the examples cited, semi- or fully automated strategies for liquid handling were incorporated to enhance throughput. Even with the recent popularity of on-line methods, off-line techniques continue to be widely employed. The key advantage to off-line methods is that sample preparation may be independently optimized from the mass spectrometer and does not contribute overhead to the LC-MS injection duty cycle. 

We now turn to the numerical solution of Equations 9.1 and 9.3. The solutions are necessarily simultaneous. The numerical techniques of Chapter 8 can be used for the simultaneous solution of Equation 9.3 with as many versions of Equation 9.1 as are necessitated by the number of components. The method of lines is unchanged except for the wall boundary condition and a new stability criterion. The marching-ahead equations (e.g., Eq. 8.31) are unchanged, but the coefficients in Tables 8.2 and 8.3 now use V(i) = Us. When the velocity profile is flat, the stability criterion of Equation 8.36... 

Chapter 8 ignored axial diffusion, and this approach would predict reactor performance like a PFR so that conversions would be generally better than in a laminar flow reactor without diffusion. However, in microscale devices, axial diffusion becomes important and must be retained in the convective diffusions equations. The method of lines ceases to be a good solution technique, and the method of false transients is preferred. Application of the false-transient technique to PDFs, both convective diffusion equations and hydrodynamic equations, is an important topic of this chapter. 

The method of lines (14) is used as the numerical technique In this method, by "finite differencing" the space variable (here axial length of reactor), the reactor is divided into a number of cells. Then the partial differential equations are converted into ordinary differential equations where time is the only independent variable. Each cell corresponds to a continuous stirred tank reactor. 

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