Nonlinear pdes

Numerical Method of Lines for Stiff Nonlinear PDEs... 

Stiff nonlinear PDEs cannot be solved using the Runge-Kutta subroutine (see chapter 2.2.5). Maple s stiff solver can he used to solve stiff nonlinear PDEs efficiently. 

The procedure developed for a single nonlinear PDE can be extended to solve coupled PDEs. Numerical method of lines provides an efficient way to solve nonlinear coupled PDEs. 

Consider the following highly coupled nonlinear PDEs[16]... 

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. 

Thus the Rayleigh problem is reduced to the solution of the ODE, (3-126), subject to the two boundary conditions, (3-127) and (3-128). When a similarity transformation works, this reduction from a PDE to an ODE is the typical outcome. Although this is a definite simplification in the present problem, the original PDE was already linear, and the existence of a similarity transformation is not essential to its solution. When similarity transformations exist for more complicated, nonlinear PDEs, however, the reduction to an ODE is often a critical simplification in the solution process. 

Although the full Navier Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u Vu were identically equal to zero or else appeared only in an equation for the crossstream pressure gradient, which was decoupled from the primary linear flow equation, as in the ID analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs. 

H. A. Stone, Partial differential equations in thin film flows in fluid dynamics Spreading droplets and rivulets, in Nonlinear PDEs in Condensed Matter and Reactive Flows, editedby H. Berestycki and Y. Pomeau, (springer-verlag, New York, 2000). 

The limiting function G t, jc) obeys the first-order nonlinear PDE ... 

The IBVP of Section 2 has two properties, which influence the solution process we have nonlinear PDEs, and the reaction rates are very different in magnitude, so that stiffness occurs. 

Loney, N.W., On using a boundary perturbation technique to linearize a system of nonlinear pde, Chem. Eng. Educ., Winter, 58, 1996. 

Lines 23 through 28 find the maximum value of the passed solution variable and then scale various probe factors, such as fctu, fctux and fctuy on lines 26 and 27, for the numerical evaluation of the functional derivatives of the defined PDE according to the maximum solution value. On line 29 an external function de-rivltreO is called to obtain values of the first derivatives of the input solution values. All of this is not necessary for a linear PDE but is included so that the code can also be used in a Newton loop for a nonlinear PDE. 

Since this is the first nonlinear PDE solved by the FE method, it is probably important to review again the technique for specifying the PDE to be solved. For this problem flie defining equation is on line 12 of Listing 13.18 and is specified as ... 

In this equation the ) (17) function assumes a similar role to the f(x,y, 17) function in the previous example. The reader is referred back to Section 12.5 for a discussion of this equation and the one dimensional time dependent solution. In the present chapter in Section 13.9 an example of linear diffusion into a two dimensional surface was presented. For that example, a triangular mesh array was developed and shown in Figure 13.15. The present example combines the nonlinear diffusion model of Section 12.5 with the FE mesh of Figure 13.15 to demonstrate a second nonlinear PDE solution using the FE approach. The reader should review this previous material as this seetion builds upon that material. 

The FE solvers developed in this chapter have made use of some of the approximate matrix solution techniques developed in the previous chapter. However, most of the code is new because of the different basic formulation of the finite element approach. In this work the method of weighted residuals has been used to formulate sets of FE node equations. This is one of the two basic methods typically used for this task. In addition, the development has been based upon flie use of basic triangular spatial elements used to cover a two dimensional space. Other more general spatial elements have been sometimes used in the FE method. Finally the development has been restricted to two spatial dimensions and with possible an additional time dimension. The code has been developed in modular form so it can be easily applied to a variety of physical problems. In keeping with the nonlinear theme of this work, the FE analysis can be applied to either linear or nonlinear PDEs. 

The developed code has been illustrated with a variety of physically based PDEs covering a range of geometries and physical disciplines. Not as much discussion on solution accuracy has been included as in previous chapters as the approach is not as amenable to a direct evaluation of solution accuracy, especially for nonlinear PDEs. 

G. Fasshauer. Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computer and Mathematics with Applications 43 (2002), 423-438. 

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