MathJax example

HLLC solver through provides a good approximation for intermediate states but is first order accurate. In this study, I used a second order scheme to simulate with the highest possible resolution. The scheme is known as MUSCL-Hancock, which aims to construct a flux that is second order extension of the first order upwind scheme. The scheme involves three steps: Data Reconstruction, Evolution and Solving a Riemann Problem. The first step involves linear extrapolation of cell centered states to obtains cell boundary values

\[U^L_i = U^n_i - \frac{1}{2} \Delta i\] \[U^R_i = U^n_i - \frac{1}{2} \Delta i\] where $ \Delta i$ is \[\Delta i = \frac{1}{2}(1+w)\Delta_{i+\frac{1}{2}} + \frac{1}{2}(1-w)\Delta_{i+\frac{1}{2}}\] \[\Delta_{i-\frac{1}{2}} = U^n_i -U^n_i+1\] \[\Delta_{i+\frac{1}{2}} = U^n_{i+1} -U^n_i\]

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The second step involves finding the intermediate states using the extrapolated values for half a time step:

\[\bar{U^L_i} = U^L_i + \frac{1}{2}\frac{\Delta t}{ \Delta x}(F(U^L_i) - F(U^R_i))\] \[\bar{U^R_i} = U^R_i + \frac{1}{2}\frac{\Delta t}{ \Delta x}(F(U^L_i) - F(U^R_i))\] The fluxes obtained from the extrapolated state vectors are boundary fluxes which is not equivalent to inter cell fluxes. The final stage involves solving a Riemann problem between $ \bar{U^L_i}$ and $\bar{U^R_i}$. In this study, HLLC solver is used. Other solver such as First Order Centered Scheme (FORCE) also exists. To make our solution total variation diminishing, I incorporate the super-bee slope limiter upon calculating $\Delta i$. The new limited slope is calculated via : \[\bar{\Delta_i} = \xi \Delta_i\] where $\xi (r)$ is known as the slope limiter and can be found from: \[\begin{cases} r = \frac{\Delta_{i-\frac{1}{2}}}{\Delta _{i+\frac{1}{2}}}\\ \xi_R(r) = \frac{ 2 \beta _{i - \frac{1}{2}}r}{1-w + (1+w)r} & \quad \beta_{i-\frac{1}{2}} = \frac{2}{1+c} \\ \end{cases}\] \[\xi_{sb}(r) = \begin{cases} 0, & \quad \quad \quad \text{if } r \leq 0 \\ 2r, & \quad \quad \quad \text{if } 0 \leq r \leq \frac{1}{2}\\ 1, & \quad \quad \quad \text{if } \frac{1}{2} \leq r \leq 1\\ min{r, \xi_R(r),2}, & \quad \quad \quad \text{if} r \geq2 \end{cases}\] c is the Courant number for the single wave present, and $w$ and $\beta$ are taking values of 0 and 1 respectively. Other slope limiter like van Leer or Minbee-type are also available. Superbee is used in this study as it is the least diffusive one, helping to maintain the sharpness of discontinuities \cite{VanLeer1984} , \cite{Quirk1994}.

References

  1. Toro, E. F. (2009), Riemann Solvers and Numerical Methods for Fluid Dynamics, thrid ed., Springer.
  2. Van Leer, B. (1976), MUSCL, a new approach to numerical gas dynamics., Computing in plasma physics and astrophysics, (January 1976).
  3. Toro, E. F., M. Spruce, and W. Speares (1994), Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4 (1), 25–34, doi:10.1007/BF01414629.
  4. Van Leer, B. (1977a), Towards the ultimate conservative difference scheme. IV. A new ap- proach to numerical convection, Journal of Computational Physics, 23(3), 276–299, doi: 10.1016/0021-9991(77)90095-X.
  5. Van Leer, B. (1977b), Towards the Ultimate Conservative Difference Scheme III. Upstream- Centered Finite-Difference Schemes for Ideal Compressible Flow, Journal of Computational Physics, 275(23), 263–275.
  6. Van Leer, B. (1979), Towards the Ultimate Conservative Difference Scheme V. A Second- Order Sequel to Godnov, Journal of Computational Physics, 32(1), 101–136, doi:10.1016/0021- 9991(79)90145-1.
  7. Van Leer, B. (1984), On the Relation Between the Upwind-Differencing Schemes of Godunov, EngquistOsher and Roe Share on, SIAM Journal on Scientific and Statistical Computing, 5(1), doi:10.1137/0905001.