Error Term for Backward or Upwind Scheme

1D Convection Equation

$$\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=0$$

Discretizing

$$\frac{\partial u}{\partial t}+c\frac{u_i-u_{i-1}}{\Delta x}=0$$

Using Taylor Expansion we have the error term can be found out (truncation error)

$$u_{i-1}=u_i-\Delta x\frac{\partial u}{\partial x}+\frac{\Delta x^2}{2!}\frac{{\partial }^{2}u}{\partial x^2}-\dots$$

where ${\Delta }_x=(u_{i-1}-u_i)$

$$\frac{u_i-u_{i-1}}{\Delta x}=\frac{\partial u}{\partial x}-\frac{\Delta x}{2!}\frac{{\partial }^{2}u}{\partial x^2}+\dots$$

The Discretized Convection Equation

$$\boxed{\frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x}=\underset{\text{Diffusion Term}}{\underbrace{c\frac{\Delta x}{2!}\frac{{\partial }^{2}u}{\partial x^2}}}-c\frac{\Delta x^2}{3!}\frac{{\partial }^{3}u}{\partial x^3}+\dots }$$

In the Continuous form, the RHS is $0$