数学代写|偏微分方程代写partial difference equations代考|МATH4335

数学代写|偏微分方程代写partial difference equations代考|BANDED LINEAR SYSTEM SOLVERS

As discussed in the preceding section, Gaussian elimination has two major shortcomings. Both shortcomings stem from the fact that the entire coefficient matrix, including all zeroes, is stored and used. As we have seen in Chapter 2, coefficient matrices, arising out of discretization of any PDE, are sparse. This is true irrespective of the type of mesh used, as will be corroborated further in Chapter 7 . Unfortunately, a computer cannot distinguish between a zero and a nonzero automatically. It treats a zero as any other real number, and multiplications by zero are as time consuming as multiplications by nonzeroes. Therefore, a better approach would be to store only the nonzeroes and the locations in the matrix where they are situated. At the very least, this will reduce the memory usage significantly. In the example discussed in the last paragraph of the preceding section, for a 64,000-node mesh, the real number storage will reduce dramatically from $64,000^2$ to at most $64,000 \times 7$, assuming a hexahedral mesh is being used along with a second-order central difference scheme. Not storing the zeroes will also improve computational efficiency dramatically. This is because multiplications by zeroes will not be performed in the first place. For example, instead of performing a row times column multiplication involving $64,000^2$ long operations, such as in the computation of $\sum_{j=1}^K A_{i, j} \phi_j$, only $7 \times 64,000$ long operations will be performed since each row will have at most seven nonzero elements. To summarize, when it comes to solution of the linear system arising out of discretization of PDEs, using the sparseness of the coefficient matrix is warranted, as it can result in significant reduction in both memory usage and computational time.

A special class of sparse matrices arises out of discretization of a PDE if a structured mesh is used: banded matrices. This has already been alluded to in Chapter 2, although detailed discussion of this matter had been deferred to the present chapter. To elucidate the matrix structures represented by the equations derived in Chapter 2 , let us consider the solution of Eq. (2.1) subject to the boundary conditions shown by Eq. (2.2).

数学代写|偏微分方程代写partial difference equations代考|ITERATIVE SOLVERS

Direct solution to a linear system of equations produces the exact numerical solution to the PDE in question. Barring truncation and round-off errors, this solution has no other error when compared with the closed-form analytical solution to the same PDE. Therein lies its strength. Unfortunately, as shown in the preceding section, direct solution is prohibitive both from a memory usage and a computational efficiency standpoint. This leaves us with only one other alternative: solving the equations iteratively.

The philosophy of an iterative solution to a linear system is best explained by a simple example. Let us consider the following $3 \times 3$ system of equations:
$$\begin{gathered} 5 x+2 y+2 z=9, \ 2 x-6 y+3 z=-1, \ x+2 y+7 z=10 . \end{gathered}$$
The exact solution to the system is $[1,1,1]$. The Gaussian elimination method, by which this exact solution is obtained, is also known as an implicit method. An implicit method of solution is one where all unknowns are treated simultaneously during the solution procedure.

The solution to the above system may also be obtained using the following iterative procedure:

2. Solve Eq. (3.17a) for $x$ and update its value using old (guessed) values of $y$ and $z$.
3. Solve Eq. (3.17b) for $y$ and update its value using old (guessed) values of $x$ and $z$.
4. Solve Eq. (3.17c) for $z$ and update its value using old (guessed) values of $x$ and $y$.
5. Replace old guess of solution by new solution.
6. Repeat steps $2-5$ (i.e., iterate) until the solution has stopped changing beyond a certain number of significant digits, to be prescribed a priori.

In each of the steps 2 through 4 , two of the three unknowns are guessed, or treated explicitly, i.e., they are treated as known quantities and transposed to the right-hand side of the equation, while the unknowns are retained on the left-hand side. The fact that some terms in the equation receive explicit treatment implies that at each step, the solution obtained is only approximate, and therefore, iterations are necessary to correct the approximation. It will be shown later that how quickly the correct solution is approached, i.e., how many iterations are needed to arrive at the correct solution, depends on how many terms are treated explicitly, or the so-called degree of explicitness. Table $3.1$ shows how the values of $[x, y, z]$ change when the above iteration procedure is executed.

偏微分方程代考

数学代写|偏微分方程代写partial difference equations代考|ITERATIVE SOLVERS

5X+2是+2和=9, 2X−6是+3和=−1, X+2是+7和=10.

1. 从对解决方案的猜测开始。
2. 求解方程。(3.17a) 对于X并使用旧的（猜测的）值更新其值是和和.
3. 求解方程。(3.17b) 对于是并使用旧的（猜测的）值更新其值X和和.
4. 求解方程。(3.17c) 为和并使用旧的（猜测的）值更新其值X和是.
5. 用新的解决方案替换旧的解决方案猜测。
6. 重复步骤2−5（即迭代）直到解决方案停止变化超过一定数量的有效数字，这是先验规定的。

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