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Fokker-Planck方程的QUICK离散格式研究
QUICK Discrete Scheme for Fokker-Planck Equation
DOI: 10.12677/AAM.2019.89177, PDF, 下载: 814  浏览: 2,108  国家自然科学基金支持

Abstract: We design a finite volume method for solving time fractional Fokker-Planck equation. The time is dispersed by L1-approximate, the space convection term is discretized by QUICK scheme, and the diffusion term is discretized by central difference. The numerical results show that the method is second-order convergent in space.

1. 研究的问题

$\frac{\partial \omega }{\partial t}=\left({k}_{\alpha }\frac{{\partial }^{2}}{\partial {x}^{2}}-\frac{\partial }{\partial x}f\right){D}_{t}^{1-\alpha }\omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\le x\le b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le T,$ (1.1)

$\omega \left(x,0\right)=\phi \left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\le x\le b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(a,t\right)={g}_{1}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(b,t\right)={g}_{2}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le T,$ (1.2)

$\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}=\left({k}_{\alpha }\frac{{\partial }^{2}}{\partial {x}^{2}}-\frac{\partial }{\partial x}f\right)\omega \left(x,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\le x\le b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le T.$ (1.3)

$|f\left(x\right)-f\left({x}^{\prime }\right)|\le Cx-{x}^{\prime },\text{\hspace{0.17em}}\text{\hspace{0.17em}}x,{x}^{\prime }\in \left[a,b\right].$ (1.4)

2. 离散

${\int }_{{x}_{i-\frac{1}{2}}}^{{x}_{i+\frac{1}{2}}}{\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}|}_{{t}_{n}}\text{d}x={k}_{\alpha }\left({\left(\frac{\partial \omega }{\partial x}\right)}_{i+\frac{1}{2},n}-{\left(\frac{\partial \omega }{\partial x}\right)}_{i-\frac{1}{2},n}\right)-\left({\left(f\omega \right)}_{i+\frac{1}{2},n}-{\left(f\omega \right)}_{i-\frac{1}{2},n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,N$ (2.1)

$\omega \left({x}_{i},{t}_{n}\right)$ 记做 ${\omega }_{i}^{n}\left(i=0,1,\cdots ,N+1;n=0,1,\cdots ,L\right)$，(2.1)式左边可以改写为：

$\begin{array}{l}{\int }_{{x}_{i-\frac{1}{2}}}^{{x}_{i+\frac{1}{2}}}{\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}|}_{{t}_{n}}\text{d}x={\left(\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}\right)}_{\left({x}_{i},{t}_{n}\right)}h-h{\gamma }_{i,n}^{\left(1\right)}\\ =\frac{h\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)}\left({\omega }_{i,n}-{\sum }_{k=1}^{n-1}\left({a}_{n-k-1}-{a}_{n-k}\right){\omega }_{i}^{k}-{a}_{n-1}{\omega }_{i}^{0}\right)-h{\gamma }_{i,n}^{\left(1\right)}-h{\gamma }_{i,n}^{\left(2\right)},\end{array}$ (2.2)

(2.3)式中第一个等式应用中矩形公式，第二个等式应用了 ${L}_{1}$ 近似，其中 ${a}_{k}={\left(k+1\right)}^{1-\alpha }-{k}^{1-\alpha }$，(见 [15] [16] )，其中误差项

${\gamma }_{i,n}^{\left(1\right)}:=\left({\left(\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}\right)}_{\left({x}_{i},{t}_{n}\right)}h-{\int }_{{x}_{i-\frac{1}{2}}}^{{x}_{i+\frac{1}{2}}}{\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}|}_{{t}_{n}}\right)/h$ (2.3)

${\gamma }_{i,n}^{\left(2\right)}:=\left(\frac{h\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)}\left({\omega }_{i,n}-{\sum }_{k=1}^{n-1}\left({a}_{n-k-1}-{a}_{n-k}\right){\omega }_{i}^{k}-{a}_{n-1}{\omega }_{i}^{0}\right)-h{\left(\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}\right)}_{\left({x}_{i},{t}_{n}\right)}\right)/h$ (2.4)

$h|{\gamma }_{i,n}^{\left(1\right)}|\le C{h}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}h|{\gamma }_{i,n}^{\left(2\right)}|\le Ch\cdot {t}^{2-\alpha },\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,N;\text{\hspace{0.17em}}n=1,\cdots ,L$

(2.1)式右侧第一项可以根据中点差分公式写作

${k}_{\alpha }\left({\left(\frac{\partial \omega }{\partial x}\right)}_{i+\frac{1}{2},n}-{\left(\frac{\partial \omega }{\partial x}\right)}_{i-\frac{1}{2},n}\right)={k}_{\alpha }\left(\stackrel{^}{{\left(\frac{\partial \omega }{\partial x}\right)}_{i+\frac{1}{2},n}}-\stackrel{^}{{\left(\frac{\partial \omega }{\partial x}\right)}_{i-\frac{1}{2},n}}\right)+h{\gamma }_{i,n}^{\left(3\right)}$ (2.5)

$\stackrel{^}{{\left(\frac{\partial \omega }{\partial x}\right)}_{i+\frac{1}{2},n}}:=\frac{{\omega }_{i+1}^{n}-{\omega }_{i}^{n}}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\cdots ,N$ (2.6)

$h|{\gamma }_{i,n}^{\left(3\right)}|\le C{h}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,N$ (2.7)

${f}_{i+\frac{1}{2}}{\omega }_{i+\frac{1}{2}}^{n}=\stackrel{˜}{{f}_{i+\frac{1}{2}}{\omega }_{i+\frac{1}{2}}^{n}}+{r}_{i+\frac{1}{2}},$ (2.8)

$\stackrel{˜}{{f}_{i+\frac{1}{2}}{\omega }_{i+\frac{1}{2}}^{n}}:={f}_{i+\frac{1}{2}}\frac{3{\omega }_{i+1}^{n}+6{\omega }_{i}^{n}-{\omega }_{i-1}^{n}}{8},$ (2.9)

${r}_{i+\frac{1}{2}}:={f}_{i+\frac{1}{2}}{\omega }_{i+\frac{1}{2}}^{n}-\stackrel{˜}{{f}_{i+\frac{1}{2}}{\omega }_{i+\frac{1}{2}}^{n}}=-\frac{1}{2}{f}_{i+\frac{1}{2}}\left(\frac{{\partial }^{3}{\omega }_{i+\frac{1}{2}}^{n}}{\partial {x}^{3}}\right){h}^{3}+ο\left({h}^{4}\right),$ (2.10)

${\left(f\omega \right)}_{i+\frac{1}{2},n}-{\left(f\omega \right)}_{i-\frac{1}{2},n}=\stackrel{˜}{{\left(f\omega \right)}_{i+\frac{1}{2},n}}-\stackrel{˜}{{\left(f\omega \right)}_{i-\frac{1}{2},n}}-h{\gamma }_{i,n}^{\left(4\right)}$ (2.11)

$h{\gamma }_{i,n}^{\left(4\right)}=\left({r}_{i-\frac{1}{2}}-{r}_{i+\frac{1}{2}}\right)$

$|{r}_{i-\frac{1}{2}}-{r}_{i+\frac{1}{2}}|\le C{h}^{3}$ .

$\begin{array}{l}\frac{h\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)}\left({\omega }_{i}^{n}-\underset{k=1}{\overset{n-1}{\sum }}\left({a}_{n-k-1}-{a}_{n-k}\right){\omega }_{i}^{k}-{a}_{n-1}{\omega }_{i}^{0}\right)\\ ={k}_{\alpha }\left(\stackrel{^}{{\left(\frac{\partial \omega }{\partial x}\right)}_{i+\frac{1}{2},n}}-\stackrel{^}{{\left(\frac{\partial \omega }{\partial x}\right)}_{i-\frac{1}{2},n}}\right)+\stackrel{˜}{{\left(f\omega \right)}_{i+\frac{1}{2},n}}-\stackrel{˜}{{\left(f\omega \right)}_{i-\frac{1}{2},n}}+h{\gamma }_{i}^{n}\end{array}$ (2.12)

$i=1,2,\cdots ,N;n=1,2,\cdots ,L$，其中 ${r}_{i}^{n}={\sum }_{j=1}^{4}{\gamma }_{i,n}^{\left(j\right)}$。我们用 ${W}_{i}^{n}$ 近似 ${\omega }_{i}^{n}$，由(2.12)式我们可以得到如下的有限体积法(FV)： $i=1,2,\cdots ,N;n=1,2,\cdots ,L$

$\begin{array}{l}\frac{h\Delta {t}^{-\alpha }}{\Gamma \left(2-\alpha \right)}\left({W}_{i}^{n}-\underset{k=1}{\overset{n-1}{\sum }}\left({a}_{n-k-1}-{a}_{n-k}\right){W}_{i}^{k}-{a}_{n-1}{W}_{i}^{0}\right)\\ ={k}_{\alpha }\left(\stackrel{^}{{\left(\frac{\partial W}{\partial x}\right)}_{i+\frac{1}{2},n}}-\stackrel{^}{{\left(\frac{\partial W}{\partial x}\right)}_{i-\frac{1}{2},n}}\right)+\stackrel{˜}{{\left(fW\right)}_{i+\frac{1}{2},n}}-\stackrel{˜}{{\left(fW\right)}_{i-\frac{1}{2},n}}+h{\gamma }_{i}^{n}\end{array}$ (2.13)

${W}_{0}^{n}={g}_{1}\left({t}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{N+1}^{n}={g}_{2}\left({t}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{i}^{0}=\phi \left({x}_{i}\right)$ (2.14)

$\left(\frac{h\Delta {t}^{-\alpha }}{\text{Γ}\left(2-\alpha \right)}I+A+B\right){W}^{n}=\frac{h\Delta {t}^{-\alpha }}{\text{Γ}\left(2-\alpha \right)}\left({\sum }_{k=1}^{n}\left({a}_{n-k-1}-{a}_{n-k}\right){W}^{k}+{a}_{n-1}{W}^{0}\right)+{d}^{n}$ (2.15)

$n=1,2,\cdots ,L$${W}^{k}={\left({W}_{1}^{k},{W}_{2}^{k},\cdots ,{W}_{N}^{k}\right)}^{\text{T}}$${d}^{n}=\left({d}_{1}^{n},{d}_{2}^{n},\cdots ,{d}_{N}^{n}\right)\in {R}^{N}$$I\in {R}^{N×N}$ 是单位矩阵。 $A=\left({a}_{ij}\right)\in {R}^{N×N}$ 是(2.13)式右侧第一项系数矩阵， $B=\left({b}_{ij}\right)\in {R}^{N×N}$ 是(2.13)式右侧第二项系数矩阵，矩阵 $A,B,{d}^{n}$ 如下：

${a}_{11}=\frac{2{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{21}=-\frac{{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{i1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ge 3；$ (2.16)

${a}_{NN}=\frac{2{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{\left(N-1\right)N}=-\frac{{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{iN}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\le N-2；$ (2.17)

${a}_{ii}=\frac{2{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{\left(i+1\right)i}={a}_{\left(i-1\right)i}=-\frac{{k}_{\alpha }}{h},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{ij}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ge j+2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\le j-2；$ (2.18)

${b}_{11}=\frac{6}{8}{f}_{\frac{3}{2}}-\frac{1}{2}{f}_{\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{21}=-\frac{1}{8}{f}_{\frac{5}{2}}-\frac{6}{8}{f}_{\frac{3}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{31}=\frac{1}{8}{f}_{\frac{5}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\ge 4$ (2.19)

${b}_{\left(N-1\right)\left(N-1\right)}=\frac{6}{8}{f}_{N-\frac{1}{2}}-\frac{3}{8}{f}_{N-\frac{3}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{\left(N-2\right)\left(N-1\right)}=\frac{3}{8}{f}_{N-\frac{3}{2}}$ (2.20)

${b}_{N\left(N-1\right)}=-\frac{1}{8}{f}_{N+\frac{1}{2}}-{f}_{N-\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{i1}=0,\text{\hspace{0.17em}}i\le N-3$ (2.21)

${b}_{NN}=\frac{6}{8}{f}_{N+\frac{1}{2}}-\frac{3}{8}{f}_{N-\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{\left(N-1\right)N}=\frac{3}{8}{f}_{N-\frac{1}{2}}$ (2.22)

${b}_{jj}=\frac{6}{8}{f}_{j+\frac{1}{2}}-\frac{3}{8}{f}_{j-\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{\left(j-1\right)j}=\frac{3}{8}{f}_{j-\frac{1}{2}}$ (2.23)

${b}_{\left(j+1\right)j}=-\frac{1}{8}{f}_{j+\frac{3}{2}}-\frac{6}{8}{f}_{j+\frac{1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{\left(j+2\right)j}=\frac{1}{8}{f}_{j+\frac{3}{2}}$ (2.24)

${d}_{1}^{n}=\left(\frac{1}{8}{f}_{\frac{3}{2}}+\frac{1}{2}{f}_{\frac{1}{2}}+\frac{{k}_{\alpha }}{h}\right){g}_{1}\left({t}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{2}^{n}=\left(-\frac{1}{8}{f}_{\frac{3}{2}}\right){g}_{1}\left(tn\right)$

${d}_{i}^{n}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=3,\cdots ,N-1，$ (2.25)

${d}_{N}^{n}=\left[-\frac{3}{8}{f}_{N+\frac{1}{2}}+\frac{{k}_{\alpha }}{h}\right]{g}_{2}\left({t}_{n}\right)。$ (2.26)

3. 数值实验及结论

$\frac{{\partial }^{\alpha }\omega }{\partial {t}^{\alpha }}=\left({k}_{\alpha }\frac{{\partial }^{2}}{\partial {x}^{2}}-\frac{\partial }{\partial x}f\left(x\right)\right)+g\left(x,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le x\le 1,\text{\hspace{0.17em}}0\le t\le 1，$ (3.1)

$g\left(x\right)=\frac{\Gamma \left(3\right)}{\Gamma \left(3-\alpha \right)}{t}^{2-\alpha }\mathrm{cos}\left(\text{π}x\right)+{k}_{\alpha }{\text{π}}^{2}{t}^{2}\mathrm{cos}\text{π}x+{t}^{2}\left[\left(1-2x\right)\mathrm{cos}\left(\text{π}x\right)-f\left(x\right)\text{π}\mathrm{sin}\left(\text{π}x\right)\right],$ (3.2)

$空间收敛阶=|\frac{\mathrm{ln}\left({‖细网格误差‖}_{1}/{‖粗网格误差‖}_{1}\right)}{\mathrm{ln}\left(细网格划分数N+1/粗网格划分数N+1\right)}|$

Table 1. Convergence rate for space with a = 0.2, L = 5000

Table 2. Convergence rate for space with a = 0.5, L = 5000

Table 3. Convergence rate for space with a = 0.8, L = 5000

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