Derive the modified differential equation corresponding to, Mechanical Engineering

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Derive the modified differential equation corresponding to

1. Generalized upwind differencing

(a) Derive the modified differential equation corresponding to the difference equation. Confirm that the method is consistent and compute the effective diffusivity.

(b) Derive the stability criteria.

2. Hopscotch method for linear three-dimensional convection-diffusion

Develop a hopscotch method for the linear convection-diffusion equation in three dimensions.

3. Fractional-step method

Write the finite-difference equations corresponding to the Crank-Nicolson discretization of the three equations.

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Mechanical Engineering: Derive the modified differential equation corresponding to

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