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Peano
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Peano
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With standard finite volume methods we assume that the solution is constant within each volume. We can construct higher order schemes in space by relaxing this assumption.
MUSCL (Monotonic Upstream-centred Scheme for Conservation Laws) schemes assume that the solution is a piecewise linear function \( u^n(x) = u_i^n + (x-x_i)\frac{s_i}{\Delta x}, x_{i-\frac{1}{2}} < x < x_{i+\frac{1}{2}} \) where \( s_i = ave(u^n_{i+1} - u^n_i, u^n_i - u^n_{i-1}) \) is the slope at volume \( i \).
The left and right face values are calculated as
\begin{eqnarray*} u^n_{i-\frac{1}{2}} = u^i_n - \frac{1}{2}s_i u^n_{i+\frac{1}{2}} = u^i_n + \frac{1}{2}s_i \end{eqnarray*}
Updating \( u_i^n \) with the flux contributions uses these face values instead of the surrounding volumes. \( u^{n + 1}_i = u^n_i + \frac{\Delta t}{\Delta x}(F_{i-\frac{1}{2}} - F_{i+\frac{1}{2}}) \) \( F_{i-\frac{1}{2}} and F_{i+\frac{1}{2}} \) are the solutions to the Riemann problem at each surrounding face. There are many approaches to solve the Riemann problem, but we use the Rusanov flux.
The MUSCL-Hancock Scheme modifies the standard MUSCL approach to calculated the flux for half a timestep later. After the face values are calculated they are updated according to the flux and source term.
1.10.0