Tridiagonal system

The sets of equations can he solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). 

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

To obtain higher-accuracy solutions for Problem 15.2, divide the interval into more subintervals. Increase n from 4 as in Problem 15.2 to 8, and decrease h from 1 to. You will generally obtain a tridiagonal system of n — 1 linear equations up to 7 in the present case. For large n, solving the system of equations with paper and pencil becomes impractical, and it is necessary to find algorithms suitable for computation by computers. 

For a series of segments where the first and last points do not match (open curve), the above equations can be re-arranged into a tridiagonal system of algebraic equations given... 

Tridiagonal systems result from Eq. (10.24a). These can be solved efficiently using the Thomas algorithm, as discussed in the last section of this chapter. 

A better alternative approach is what will be called the Rudolph method [476], after the person who introduced it into electrochemical simulation. It was known before 1991 under various names, notably block-tridiagonal [280,412,470,471,528,570]. This comes from the fact that if one lumps the large matrix into a matrix of smaller matrices and vectors, the result is a tridiagonal system that is amenable to more efficient methods of solution. In the present context, we define some vectors... 

Thus far, this looks just like the Thomas algorithm for the tridiagonal system, as described above in Sect. 8.3. Prom here on, however, the processes diverge. We need to keep both substitutions for C N and C N 1 and use them in the third-last equation, which contains both. This process is continued backwards, reducing all equations with four unknowns to new ones with just two unknowns. The expressions resulting from this are the following ... 

The second example from the general system (284) is of direct relevance to the Lanczos continued fraction (LCF). The Lanczos inhomogeneous tridiagonal system of linear equations can be identified from Eq. (284) by specifying D a and ... 

The temperatures at time tk on the right hand side of (2.268) are known the three unknown temperatures at time tk+1 on the left hand side have to be calculated. The difference equation (2.268) yields a system of linear equations with i = 1,2,... n. The main diagonal of the coefficient matrix contains the elements (2 + 2M) the sub- and superdiagonals are made up of the elements (—M) all other coefficients are zero. In this tridiagonal system, the first equation (i = 1) cannot contain the term —Mi q+1 and likewise, in the last equation (i = n)... 

If the temperatures at the boundaries are given, the grid is chosen such that x = x0 and x = xn+1 coincide with the two boundaries. This means that -dk, do+1 and are always known, and the first equation of the tridiagonal system... 

When considering the heat transfer condition (2.253) we lay the grid out as in Fig. 2.45, namely so that the boundary coincides with x1 or xn, when (2.253) is stipulated at the left or right boundary. Once again i9q and q+1 are eliminated from the first equation of the tridiagonal system (2.268), this time using (2.254). This yields... 

The tridiagonal system for each time step, which has to be solved consists of five equations and has the form... 

This equation is to be formulated for all grid points (i, j). A system of linear equations for the unknown temperatures at time tk+l, that has to be solved for every time step, is obtained. Each equation contains five unknowns, only the temperature at the previous time tk is known. A good solution method has been presented by P.W. Peaceman and H.H. Rachford [2.69]. It is known as the alternating-direction implicit procedure (ADIP). Here, instead of the equation system (2.305) two tridiagonal systems are solved, through which the computation time is reduced, see also [2.53]. 

The discretisation of the heat conduction equation can also be undertaken for three-dimensional temperature fields, and this is left to the reader to attempt. The stability condition (2.304) is tightened for the explicit difference formula which means time steps even smaller than those for planar problems. The system of equations of the implicit difference method cannot be solved by applying the ADIP-method, because it is unstable in three dimensions. Instead a similar method introduced by J. Douglas and H.H. Rachford [2.71], [2.72], is used, that is stable and still leads to tridiagonal systems. Unfortunately the discretisation error using this method is greater than that from ADIP, see also [2.53]. 

Coefficients a, a2 and b are obtained by the Fourier analysis and the relatively rapid solution of the resulting tridiagonal system of equations, due to the implicit nature of (2.31). A typical set is a = 22, a% = 1, and b = 24. To comprehend their function, let us observe Figure 2.2 that assumes the computation of dHy/dx and dEz/dx at i = 0. For the first case, constraint Ey = Ez = Hx = 0 at i = 0 indicates that dHy/dx (likewise for all H derivatives) must also be zero. In the second case, to calculate dEz/dx at i = one needs its values at i = —, . Nonetheless, point i = — is outside the domain and to find a reliable value for the tridiagonal matrix, the explicit, sixth-order central-difference scheme is selected... 

Hindmarsh, A. C. Solution of Block-Tridiagonal Systems of Linear Algebraic Equations Lawrence Livermore National Laboratory Report UCID-30150, 1977. 

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