多分辨率WCNS格式记录

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已于 2024-07-02 20:17:02 修改

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文章标签： matlab 算法

于 2024-07-02 20:11:19 首次发布

本文链接：https://blog.youkuaiyun.com/cccczhh/article/details/140134383

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参考文献：

[1] Wang Z, Zhu J, Wang C, et al. An efficient hybrid multi-resolution WCNS scheme for solving compressible flows[J]. Journal of Computational Physics, 2023, 477: 111877.

[2] Zhang H, Wang G, Zhang F. A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws[J]. International Journal of Computational Fluid Dynamics, 2020, 34(3): 187-203.

问题1：利用WCNS格式，重新计算文献1中方程(3.20)与(3.21)中的光滑因子，并给出具体的表达式。

答：

${\beta _1}{\rm{ = }}\frac{1}{{72}}\left( \begin{array}{l} 91u_{i - 2}^2 + 1384u_{i - 1}^2 + 3042u_i^2 + 1384u_{i + 1}^2 + 91u_{i + 2}^2\\ - 672{u_{i - 2}}{u_{i - 1}} + 894{u_{i - 2}}{u_i} - 512{u_{i - 2}}{u_{i + 1}} + 108{u_{i - 2}}{u_{i + 2}}\\ - 3936{u_{i - 1}}{u_i} + 2352{u_{i - 1}}{u_{i + 1}} - 512{u_{i - 1}}{u_{i + 2}}\\ - 3936{u_i}{u_{i + 1}} + 894{u_i}{u_{i + 2}} - 672{u_{i + 1}}{u_{i + 2}} \end{array} \right),$

${\beta _2}{\rm{ = }}{\left( {{u_{j - 1}} - 2{u_j} + {u_{j + 1}}} \right)^2} + \frac{1}{4}{\left( { - {u_{j - 1}} + {u_{j + 1}}} \right)^2},$

${\beta _3}=\min[(u_{j-1}-u_{j})^2,(u_{j}-u_{j+1})^2].$

对应的Mathematica代码如下，输出的 $\beta_1$ 是没有展开的， $\beta_3$ 是照抄文献1的。

Clear["Global'*"]
r = 5;
(*u_{i+1/2}^{-}*)
p[1, x_] = (u[-2] - 4*u[-1] + 6*u[0] - 4*u[1] + u[2])/24*(x)^4 +
   (-u[-2] + 2*u[-1] - 2*u[1] + u[2])/12*x^3 +
   (-u[-2] + 16*u[-1] - 30*u[0] + 16*u[1] - u[2])/24*x^2 +
   (u[-2] - 8*u[-1] + 8*u[1] - u[2])/12*x + u[0];

p[2, x_] = (u[-1] - 2*u[0] + u[1])/2*x^2 + (u[1] - u[-1])/2*x + u[0];

p[3, x_] = u[0];


(*Do[Print["IS",k,"=",FullSimplify[Sum[Integrate[(D[p[k,x],{x,i}])^2,{\
x,-1/2,1/2}],{i,1,r-1}]]],{k,1,2}]*)

Do[Print["IS", k, "=", 
  FullSimplify[Sum[(D[p[k, x], {x, i}])^2, {i, 1, r - 1}]] /. 
   x -> 0], {k, 1, 2}]