Question

Part A: Consider the Poisson equation $\nabla^2 u = f(x, y)$ (1) for $0 \le x \le 1$, $0 \le y \le 1$. Figure 1: Discretization of the second derivative on a uniform mesh. 1. Discretize (1) on a uniform mesh using the following approximation for the second derivatives (see Fig. 1): $u_{xx}(x_i) = \frac{u(x_{i+1}) - u(x_i)}{x_{i+1} - x_i} - \frac{u(x_i) - u(x_{i-1})}{x_i - x_{i-1}} \approx \frac{1}{x_{i+1} - x_{i-1}} \left[ \frac{u(x_{i+1}) - u(x_i)}{x_{i+1} - x_i} - \frac{u(x_i) - u(x_{i-1})}{x_i - x_{i-1}} \right]$ (2) This expression is second-order accurate on a uniform grid. Use an analogous expression for the y-derivatives. Use the cell-centered grid arrangement shown in Fig. 2. The ghost points (shaded) can be used to evaluate the boundary condition. Figure 2: Grid system for the solution of the Poisson equation. The shaded points are ghost points.

          Part A:
Consider the Poisson equation
$\nabla^2 u = f(x, y)$
(1)
for $0 \le x \le 1$, $0 \le y \le 1$.

Figure 1: Discretization of the second derivative on a uniform mesh.

1. Discretize (1) on a uniform mesh using the following approximation for the second derivatives (see
Fig. 1):

$u_{xx}(x_i) = \frac{u(x_{i+1}) - u(x_i)}{x_{i+1} - x_i} - \frac{u(x_i) - u(x_{i-1})}{x_i - x_{i-1}} \approx \frac{1}{x_{i+1} - x_{i-1}} \left[ \frac{u(x_{i+1}) - u(x_i)}{x_{i+1} - x_i} - \frac{u(x_i) - u(x_{i-1})}{x_i - x_{i-1}} \right]$
(2)

This expression is second-order accurate on a uniform grid. Use an analogous expression for the
y-derivatives. Use the cell-centered grid arrangement shown in Fig. 2. The ghost points (shaded) can
be used to evaluate the boundary condition.

Figure 2: Grid system for the solution of the Poisson equation. The shaded points are ghost points.

Part A:
Consider the Poisson equation
∇^2 u = f(x, y)
(1)
for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

Figure 1: Discretization of the second derivative on a uniform mesh.

1. Discretize (1) on a uniform mesh using the following approximation for the second derivatives (see
Fig. 1):

uxx(xi) = (u(xi+1) - u(xi))/(xi+1 - xi) - (u(xi) - u(xi-1))/(xi - xi-1)≈(1)/(xi+1 - xi-1)[ (u(xi+1) - u(xi))/(xi+1 - xi) - (u(xi) - u(xi-1))/(xi - xi-1)]
(2)

This expression is second-order accurate on a uniform grid. Use an analogous expression for the
y-derivatives. Use the cell-centered grid arrangement shown in Fig. 2. The ghost points (shaded) can
be used to evaluate the boundary condition.

Figure 2: Grid system for the solution of the Poisson equation. The shaded points are ghost points.

Added by Brandon O.

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