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MusclHancock

-              v22
+              v23
 for each element of u, where f is the numerical flux. For the 1st order Godunov scheme studied previously, f was taken to simply be the physical flux of 2 adjacent cells, evaluated using the solution to the Local Riemann Problem at the intercell boundary. We have seen various approximation methods to this Godonov scheme, some which estimate the solution of the Riemann problem itself along these intercell boundaries, and others that instead approximate the numerical flux at these boundaries. Next, we will see how the MH scheme produces a 2nd-order accurate solution to the problem. The MH scheme proceeds as follows, being broken down into 3 distinct steps.
+.
+. 'Data reconstruction' step - this is the first step of the MH scheme, and is responsible for providing higher order accuracy by fitting a piece-wise linear function to the initial piece-wise constant data. Note that this steps maintains conservation. Such a piece-wise linear function is of the form:
+[[latex($u_i(x) = u_i^n + (x - x_i) \frac{\triangle _i }{\triangle x}, x \in [0, \triangle x]$)]]
 At this point, we are left with a higher order accurate code, but not one that is free of spurious oscillations near large gradients. To circumvent this, we need to add a TVD measure.