Numerical Implementations for 2D Lid-Driven Cavity Flow in Stream Function Formulation

Table 1

Second-order central difference approximations.

𝛿𝑥𝜓𝑖,𝑗	𝜓𝑖+1,𝑗−𝜓𝑖−1,𝑗/2ℎ𝑥
𝛿𝑦𝜓𝑖,𝑗	𝜓𝑖,𝑗+1−𝜓𝑖,𝑗−1/2ℎ𝑦
𝛿𝑥2𝜓𝑖,𝑗	𝜓𝑖+1,𝑗−2𝜓𝑖,𝑗+𝜓𝑖−1,𝑗/ℎ^2𝑥
𝛿𝑦2𝜓𝑖,𝑗	𝜓𝑖,𝑗+1−2𝜓𝑖,𝑗+𝜓𝑖,𝑗−1/ℎ^2𝑦
𝛿𝑥3𝜓𝑖,𝑗	𝜓𝑖+2,𝑗−2𝜓𝑖+1,𝑗+2𝜓𝑖−1,𝑗−𝜓𝑖−2,𝑗/2ℎ^3𝑥
𝛿𝑦3𝜓𝑖,𝑗	𝜓𝑖,𝑗+2−2𝜓𝑖,𝑗+1+2𝜓𝑖,𝑗−1−𝜓𝑖,𝑗−2/2ℎ^3𝑦
𝛿𝑥2𝑦𝜓𝑖,𝑗	𝜓𝑖+1,𝑗+1−𝜓𝑖+1,𝑗−1−2𝜓𝑖,𝑗+1−𝜓𝑖,𝑗−1+𝜓𝑖−1,𝑗+1−𝜓𝑖−1,𝑗−1/2ℎ𝑦ℎ^2𝑥
𝛿𝑥𝑦2𝜓𝑖,𝑗	𝜓𝑖+1,𝑗+1−𝜓𝑖−1,𝑗+1−2𝜓𝑖+1,𝑗−𝜓𝑖−1,𝑗+𝜓𝑖+1,𝑗−1−𝜓𝑖−1,𝑗−1/2ℎ𝑥ℎ^2𝑦
𝛿𝑥4𝜓𝑖,𝑗	𝜓𝑖+2,𝑗−4𝜓𝑖+1,𝑗+6𝜓𝑖,𝑗−4𝜓𝑖−1,𝑗+𝜓𝑖−2,𝑗/ℎ^4𝑥
𝛿𝑦4𝜓𝑖,𝑗	𝜓𝑖,𝑗+2−4𝜓𝑖,𝑗+1+6𝜓𝑖,𝑗−4𝜓𝑖,𝑗−1+𝜓𝑖,𝑗−2/ℎ^4𝑦
𝛿𝑥2𝑦2𝜓𝑖,𝑗	𝜓𝑖+1,𝑗+1−2𝜓𝑖+1,𝑗+𝜓𝑖+1,𝑗−1−2𝜓𝑖,𝑗+1−2𝜓𝑖,𝑗+𝜓𝑖,𝑗−1+𝜓𝑖−1,𝑗+1−2𝜓𝑖−1,𝑗+𝜓𝑖−1,𝑗−1/ℎ^2𝑥ℎ^2𝑦