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MusclHancock

-              v38
+              v39
 where u is the vector of conserved variables (u=u(x,t)), and f is the flux function. Adhering to the IVBP known as the Riemann Problem (i.e. constant left and right data states separated by an initial discontinuity), we solve this system of equations using the fully discrete, explicit, conservative formula:
 [[latex($ u_i^ {n+1} = u_i ^n + \frac{\triangle t}{\triangle x}[f_{i-1/2} - f_{i+1/2}] *$)]]
+[[latex($ u_i^ {n+1} = u_i ^n + \frac{\triangle t}{\triangle x}[f_{i-1/2} - f_{i+1/2}] (*)$)]]
 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.
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 = Results =
+As you can see here, the MH scheme + Superbee slope limiter + Roe solver produce highly non-diffusive results. However, there still are some small scale spurious oscillations near the boundary of the square wave.
 [[Image(MusclRho.png, 35%)]], [[Image(ToroMusclRho.png, 20%)]]