# 数学代写|偏微分方程代写partial difference equations代考|AMATH353

## 数学代写|偏微分方程代写partial difference equations代考|COORDINATE TRANSFORMATION TO CURVILINEAR COORDINATES

Until this point, the derivations of finite difference approximations have been limited to a regular Cartesian mesh. As shown in Section 2.5, if the nodes are not placed along the Cartesian grid lines, Taylor series expansions result in nonzero crossderivative containing terms. Hence, derivation of finite difference approximations becomesquite complicated and tedious. Tothe best of thisauthor’sknowledge, body-fitted curvilinear meshes have their early roots in computational fluid dynamics, particularly for aerospace applications involving external flow. This happened at a time when researchers realized that the only accurate way to compute velocity distributions and, subsequently, drag and lift on an airfoil, is to construct a mesh that aligned with the profile of the airfoil. Such ideas were driven by conformal maps (angle preserving coordinate transformations) introduced by the Russian aerodynamicist Zhukovsky (transliterated into Joukowsky in the western scientific literature in the English language) for the analysis of airfoils, in which he transformed a circle into the fish-like shape of an airfoil. Since the early 1980s, the use of body-fitted structured meshes has seen tremendous growth and still continues to find prolific usage in numerical solution of PDEs, although it is slowly giving way to unstructured meshes, especially in the context of the finite volume method.

The basic idea behind a coordinate transformation is to transform the governing differential equation written in Cartesian coordinates to a new coordinate system. This new coordinate system may have axes that are either straight lines or curves. For the general case of curved axes, the new coordinate is known as a curvilinear coordinate system. When these curved axes align exactly with the outer contour of the computational domain (or body), the resulting mesh is called a body-fitted mesh or body-fitted grid. The main difference between the Cartesian coordinate system and a curvilinear coordinate system is that in the Cartesian coordinate system the unit vectors (or basis vectors) are global, while in a curvilinear system, the basis vectors are local (locally tangential to the curve) and are known as covariant basis vectors. Figure $2.11$ shows a Cartesian coordinate system $\left(x_1, x_2, x_3\right)$ and a general curvilinear coordinate system $\left(\xi_1, \xi_2, \xi_3\right)$. In the case of the Cartesian coordinate system, the unit vectors $\hat{x}_1, \hat{x}_2$, and $\hat{x}_3$ are global, while in the case of the curvilinear coordinate system, the unit vectors $\hat{\xi}_1, \hat{\xi}_2$, and $\hat{\xi}_3$ are local.

In general, the transformation from $\left(x_1, x_2, x_3\right)$ to $\left(\xi_1, \xi_2, \xi_3\right)$ is the so-called forward transformation and may be written as
\begin{aligned} &x_1=x_1\left(\xi_1, \xi_2, \xi_3\right) \ &x_2=x_2\left(\xi_1, \xi_2, \xi_3\right) . \ &x_3=x_3\left(\xi_1, \xi_2, \xi_3\right) \end{aligned}
In most cases, the forward transformation can be written in explicit functional form. The opposite transformation, the so-called backward transformation, cannot usually be written in explicit form. It usually appears as an implicit relationship.

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

Direct solution to a set of linear algebraic equations is obtained by the method of substitution in which equations are successively substituted into other equations to reduce the number of unknowns until, finally, one unknown remains. If performed on a computer, the solution obtained by this method has no errors other than round-off errors. Hence, the solution obtained by a direct solver is often referred to as the exact numerical solution. It is termed “numerical” because the governing PDE still has to be discretized and solved numerically. It is termed “exact” because the algebraic equations resulting from discretization of the PDE are solved exactly. Depending on the structure of the coefficient matrix – whether it is full or sparse or banded – the number of substitutions needed may vary. In this subsection, we discuss two algorithms based on the general philosophy of substitution. The first of these methods, known as Gaussian elimination (or Gauss-Jordan elimination), is the most general method to solve a system of linear equations directly without any assumption with regard to the nature of the coefficient matrix, i.e., the method allows for the fact that the coefficient matrix may be full. Later in this section, we discuss a class of direct solvers in which the coefficient matrix is banded.

The process of solving a set of linear algebraic equations by Gaussian elimination involves two main steps: forward elimination and backward substitution. To understand each of theses steps, let us consider a system of $K$ linear algebraic equations of the general form shown in Eq. (2.35), which, in expanded form, may be written as
$$\begin{array}{ccccc} A_{1,1} \phi_1 & +A_{1,2} \phi_2 & +\ldots & +A_{1, K} \phi_K & =Q_1 \ A_{2,1} \phi_1 & +A_{2,2} \phi_2 & +\ldots & +A_{2, K} \phi_K & =Q_2 \ \vdots & \vdots & & \vdots & \ A_{i, 1} \phi_1 & +A_{i, 2} \phi_2 & +\ldots & +A_{i, K} \phi_K & =Q_i \ \vdots & \vdots & & \vdots & \ A_{K, 1} \phi_1 & +A_{K, 2} \phi_2 & +\ldots & +A_{K, K} \phi_K & =Q_K \end{array}$$
where $A_{i, j}$ are the elements of the coefficient matrix $[A], Q_i$ are the elements of the right-hand side vector $[Q]$, and $\phi_i$ are the unknowns. In the forward elimination step, we start from the first (topmost) equation, and express $\phi_i$ in terms of all the other $\phi$ ‘s. This yields the following equation:
$$\phi_1=\frac{Q_1}{A_{1,1}}-\frac{A_{1,2}}{A_{1,1}} \phi_2-\frac{A_{1,3}}{A_{1,1}} \phi_3-\ldots-\frac{A_{1, K}}{A_{1,1}} \phi_K .$$
Next, we substitute Eq. (3.2) into each of the equations in Eq. (3.1) except the first (topmost) one.

# 偏微分方程代考

## 数学代写|偏微分方程代写partial difference equations代考|COORDINATE TRANSFORMATION TO CURVILINEAR COORDINATES

$$x_1=x_1\left(\xi_1, \xi_2, \xi_3\right) \quad x_2=x_2\left(\xi_1, \xi_2, \xi_3\right) . x_3=x_3\left(\xi_1, \xi_2, \xi_3\right)$$

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

$$\phi_1=\frac{Q_1}{A_{1,1}}-\frac{A_{1,2}}{A_{1,1}} \phi_2-\frac{A_{1,3}}{A_{1,1}} \phi_3-\ldots-\frac{A_{1, K}}{A_{1,1}} \phi_K .$$

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