三阶能量稳定WENO 格式
- Third order WENO
Consider the hyperbolic conservation law $$u_t+f(u)_x=0 $$
Assume that \(f'(u)>0\), the third WENO scheme is given by
\[\begin{align}
\frac{d u_j}{dt}+\frac{f_{j+1/2} -f_{j-1/2} }{\Delta} &=0\\
f_{j+1/2} &=w^0_{j+1/2} f^{(0)}_{j+1/2} +w^1_{j+1/2} f^{(1)}_{j+1/2} \\
f^{(0)}_{j+1/2} &=1/2 f(u_j)+1/2f(u_{j+1}) \\
f^{(1)}_{j+1/2} &=-1/2 f(u_{j-1})+3/2 f(u_j)
\end{align}
\]
The classical weight functions are given by
$$
\begin{align}
w^k_{j+1/2} &=\frac{\alpha_k }{ \alpha_0 +\alpha_1}, k=0,1 \\
\alpha_k &=\frac{d_k}{(\varepsilon +\beta_k)^2} \\
d_0&=2/3 \\
d_1&=1/3\\
\beta_0 &=(u_{j+1}-u_j )^2 \\
\beta_1 &=(u_j-u_{j+1})^2
\end{align}
$$
It is well known that the standard weight functions make the corresponding WENO scheme too dissipative. Furthermore, the conventional third-order WENO scheme presented above may locally degenerate to second order near local extrema. These properties of the classical weight functions indicate that new weight functions are needed to recover the design order of convergence near the critical points.
- ESWENO
\[\begin{align}
\alpha_k&=d_k(1+\frac{\tau}{\epsilon +\beta_k})\\
\tau &=(u_{j+1}-2u_{j}+u_{j-1})^2
\end{align}
\]
参考文献
N.K. Yamaleev, M.H. Carpenter / Journal of Computational Physics 228 (2009) 3025–3047
