Numerical method of lines

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

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

W E Shicsscr, The numerical method of lines, Academic Press San Diego, 1991... 

The numerical Method of Lines as implemented in the routine NDSolve of the Mathematica system deals with system (32) by employing the default fourth order finite difference discretization in the spatial variable Z, and creating a much larger coupled system of ordinnary equations for the transformed dimensionless temperature evaluated on the knots of the created mesh. This resulting system is internally solved (still inside NDSolve routine) with Gear s method for stiff ODE systems. Once numerical results have been obtained and automatically interpolated by NDSolve, one can apply the inverse expression (31.b) to obtain the full dimensionless temperature field. 

Numerical Method of Lines for First Order Hyperbolic... 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab or the material in which the experiment is performed. This process is usually time varying and eventually reaches a steady state. This process is represented by parabolic partial differential equations with known initial conditions and boundary conditions at two ends. Both linear and nonlinear parabolic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear parabolic partial differential equations and numerical solutions for nonlinear parabolic partial differential equations based on the numerical method of lines. 

The numerical method of lines[l] [3] [4] [2] (Schiesser and Silebi, 1997 Cutlip and Shacham, 1999 Taylor 1999 Constantinides and Mostoufi, 1999) involves converting the governing equation (equation (5.48)) to a system of coupled ODEs in time by applying finite difference approximations for the spatial derivatives... 

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. 

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. 

Solve this linear problem using numerical method of lines for Pe = 1, 10. How many node points are needed for obtaining three digits accuracy if average concentration at t = 1 is used to verify convergence ... 

---