I try to solve the one-dimensional convection diffusion reaction equation numerically by use of Crank-Nicholson scheme, its partial differential equation form as: this (click)

In the Crank-Nicholson scheme, the partial derivatives are approximated as: this (click)

Since the results by the Crank-Nicholson scheme is oscillating I need to switch to the implicit Backward-Euler scheme but could not find approximations for the partial derivatives.

Would you please give me the difference expressions for ∂²u/∂x², ∂u/∂x, ∂u/∂t, and u for the Backward Euler Scheme?

For more details: pg. 10 in reference


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