Changes between Version 37 and Version 38 of u/erica/MusclHancock
 Timestamp:
 06/18/13 13:23:35 (11 years ago)
Legend:
 Unmodified
 Added
 Removed
 Modified

u/erica/MusclHancock
v37 v38 28 28 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: 29 29 30 [[latex($ u_i^ {n+1} = u_i ^n + \frac{\triangle t}{\triangle x}[f_{i1/2}  f_{i+1/2}] $)]]30 [[latex($ u_i^ {n+1} = u_i ^n + \frac{\triangle t}{\triangle x}[f_{i1/2}  f_{i+1/2}] *$)]] 31 31 32 32 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 2ndorder accurate solution to the problem. The MH scheme proceeds as follows, being broken down into 3 distinct steps. … … 101 101 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. There are some different ways of doing this, i.e. limited slopes, fluxlimiters, and slope limiters. Using one of these methods, you would replace the slope function in steps 1 and 2 with a TVD version of the slope. This leads to a scheme that is both higher order accurate as well as minimizes spurious oscillations. 102 102 103 To summarize, the MH scheme provides a means to solve for the numerical flux of the conservative update formula for cell i (*). This flux is just the regular Godunov flux, but now evaluated using the solution of the LRP consisting of left and right states that fit a piecewise linear function. This linear function is chosen to both a) increase the order of accuracy, and b) satisfy TVD conditions. In the end we are left with a numerical solution that is both less dispersive in regions of smooth flow and less oscillatory near sharp gradients. 104 103 105 = Results = 104 106