Let the 1D burgers equation be defined by $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = v \frac{\partial^2 u}{\partial x^2}$ where $v = \frac{1}{Re}$ where $Re$ is the number of Reynolds and $a$ is the velocity component of a fluid in the $x$-axis direction. To discretize the terms of the equation at the point $P=(i,j)$ of the computational grid
- advanced or progressive difference for the time term $$\frac{\partial u}{\partial t} = \frac{u_{i,j+1}-u_{i,j}}{k}$$
- central difference for the viscous term $$\frac{\partial^2 u}{\partial x^2} = \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2}$$
- upwind strategy for the convective term $$CONV(u)=a\frac{\partial u}{\partial x}=\frac{\partial (\frac{1}{2}au)}{\partial x}=\frac{1}{2} \frac{\left( a_{i+\frac{1}{2},j}*u_{i+\frac{1}{2},j}-a_{i-\frac{1}{2},j}*u_{i-\frac{1}{2},j}\right) }{h}$$ in which $$a_{i+\frac{1}{2},j}=\frac{1}{2} \left( u_{i+1,j}+u_{i,j} \right)$$ y $$a_{i-\frac{1}{2},j}=\frac{1}{2} \left( u_{i-1,j}+u_{i,j} \right)$$ by the FOU scheme is found $u_{i+\frac{1}{2},j}$ and $u_{i-\frac{1}{2},j}$ $$a_{i+1/2,j} > 0 \rightarrow u_{i+1/2,j}=u_{i,j}$$ $$a_{i+1/2,j} \leq 0 \rightarrow u_{i+1/2,j}=u_{i+1,j}$$ $$a_{i-1/2,j} > 0 \rightarrow u_{i-1/2,j}=u_{i-1,j}$$ $$a_{i-1/2,j} \leq 0 \rightarrow u_{i-1/2,j}=u_{i,j}$$ then the explicit numerical method that is determined to solve the burgers equation is given by $$\frac{u_{i,j+1}-u_{i,j}}{k} + \frac{1}{2} \frac{\left( a_{i+\frac{1}{2},j}*u_{i+\frac{1}{2},j}-a_{i-\frac{1}{2},j}*u_{i-\frac{1}{2},j}\right) }{h} = v \frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{h^2}$$
$$u_{i,j+1}=-kCONV(u)+\frac{kv}{h^2}u_{i+1,j}+\left(1-2\frac{kv}{h^2}\right)u_{i,j}+\frac{kv}{h^2}u_{i-1,j}$$
Is this correct or am I wrong?