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I came across this paper comparing various boundary conditions. I am particularly interested to understand how to obtain the Reynolds boundary conditions (refer to equation 28).

$$\left( \frac{1}{c} \frac{\partial}{\partial t} -\frac{\partial}{\partial x} \right) \left( \frac{p}{c} \frac{\partial}{\partial t} -\frac{\partial}{\partial x} \right)=0 $$ at the boundary $x=-a$.

I looked for the discretized equations in the original paper which are given as follows

$$ u_{1,n}^{j+1} = u_{1,n}^j + u_{2,n}^{j} - u_{2,n}^{j-1} + \frac{c \Delta t}{h} \left( u_{2,n}^j - u_{1,n}^{j} - u_{3,n}^{j-1} - u_{2,n}^{j-1} \right) $$

I tried to backtrack the equation as follow

$$ u_{1,n}^{j+1} - u_{1,n}^j - u_{2,n}^{j} + u_{2,n}^{j-1} = \frac{c \Delta t}{h} \left( u_{2,n}^j - u_{1,n}^{j} - u_{3,n}^{j-1} - u_{2,n}^{j-1} \right) $$ My observation predicts a mixed derivative $\frac{\partial^2 }{\partial x \partial t}$ on both sides. which is not consistent with the formulation.

Can anyone help me to understand how these equations are derived?

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