## 临界Schrödinger映射非齐次初边值问题的有限差分格式

1 南京审计大学应用数学系 南京 211815

2 曲阜师范大学数学科学学院 山东曲阜 273165

3 西安工程大学理学院 西安 710048

## Finite Difference Scheme for the Nonhomogeneous Initial Boundary Value Problem of Critical Schrödinger Map

Deng Haiyun,1, Liu Hui,2, Song Wenjing,3

1 Department of Applied Mathematics, Nanjing Audit University, Nanjing 211815

2 School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165

3 College of Science, Xi'an Polytechnic University, Xi'an 710048

 基金资助: 国家自然科学基金.  12001276国家自然科学基金.  12001275国家自然科学基金.  12071219国家自然科学基金.  11901342国家自然科学基金.  12001415山东省自然科学基金.  ZR2018QA002中国博士后科学基金.  2019M652350

 Fund supported: the NSFC.  12001276the NSFC.  12001275the NSFC.  12071219the NSFC.  11901342the NSFC.  12001415the NSF of Shandong Province.  ZR2018QA002the China Postdoctoral Science Foundation.  2019M652350

Abstract

In this paper, we study finite difference scheme for the nonhomgeneous initial boundary value problem of critical Schrödinger map in two-dimensional space. We get the convergence and stability of the difference scheme. At the same time, we prove that this difference scheme has good effectiveness and stability by numerical experiments.

Keywords： Schrödinger map ; Difference scheme ; Convergence ; Stability

Deng Haiyun, Liu Hui, Song Wenjing. Finite Difference Scheme for the Nonhomogeneous Initial Boundary Value Problem of Critical Schrödinger Map. Acta Mathematica Scientia[J], 2021, 41(5): 1311-1322 doi:

## 2 预备知识

$$$u_t(x, y, t) = u(x, y, t)\times\Delta u(x, y, t), \; (x, y)\in\Omega, t\in [0, T],$$$

$$$u|_{t = 0} = \varphi(x, y), \; (x, y)\in\Omega,$$$

$$$u_{0j} = u_{i0} = g_0(t), u_{Jj} = u_{iJ} = g_1(t),$$$

$$$g_0(t), g_1(t)\in C^{2k+1}([0, T], S^2), \varphi(x, y)\in H^{2k+1}(\Omega, S^2),$$$

$\begin{eqnarray} \frac{u_{ij}^{n + 1} - u_{ij}^n}{\tau } = \frac{u_{ij}^{n + 1} + u_{ij}^n}{2} \times \frac{(u_{i + 1, j}^n - 2u_{ij}^n + u_{i - 1, j}^n) + (u_{i, j + 1}^n - 2u_{ij}^n + u_{i, j - 1}^n)}{h^2}, \end{eqnarray}$

$$$u_{ij}^0 = \varphi ({x_i}, {y_j}), (i, j = 1, 2, \cdots , J - 1),$$$

$$$u_{0j}^n = u_{i0}^n = {g_0}(n\tau ), u_{Jj}^n = u_{iJ}^n = {g_1}(n\tau ), (n = 1, 2, \cdots , N).$$$

$$$\frac{{u_{ij}^{n + 1} - u_{ij}^n}}{\tau } = \frac{{u_{ij}^{n + 1} + u_{ij}^n}}{2} \times \frac{{{\Delta _ + }{\Delta _ - }u_{ij}^n}}{{{h^2}}},$$$

$$$u_{ij}^{n + 1} = u_{ij}^n + \frac{\tau }{{2{h^2}}}(u_{ij}^{n + 1} + u_{ij}^n) \times {\Delta _ + }{\Delta _ - }{u_{ij}}.$$$

## 3 离散解的存在性

由(2.9)式, 我们定义一个离散函数$\Phi_{ij}$满足如下关系:

$$${\Phi _{ij}} = u_{ij}^n + \mu \frac{\tau }{{2{h^2}}}({\Phi _{ij}} + u_{ij}^n) \times {\Delta _ + }{\Delta _ - }{u_{ij}}, (i, j = 1, 2, \cdots , J - 1),$$$

$$$u_{0j}^n = u_{i0}^n = g_0(n\tau), u_{Jj}^n = u_{iJ}^n = g_1(n\tau), (n = 1, 2, \cdots , N),$$$

$$$\sum\limits_{i, j = 1}^{J - 1} {({\Phi _{ij}}, {\Phi _{ij}})} = \sum\limits_{i, j = 1}^{J - 1} {(u_{ij}^n, {\Phi _{ij}})} + \mu \frac{\tau }{{2{h^2}}}\sum\limits_{i, j = 1}^{J - 1} {(u_{ij}^n \times {\Delta _ + }{\Delta _ - }{u_{ij}}, {\Phi _{ij}})},$$$

$$$\frac{1}{h}||{\Phi _{ij}}||_{L^2}^2 = \sum\limits_{i, j = 1}^{J - 1} {(u_{ij}^n, {\Phi _{ij}})} + \mu \frac{\tau }{{2{h^2}}}\sum\limits_{i, j = 1}^{J - 1} {(u_{ij}^n \times {\Delta _ + }{\Delta _ - }{u_{ij}}, {\Phi _{ij}})} ,$$$

由(2.8)及(3.7)式可知，$e_{ij}^n$满足如下关系式

$\begin{eqnarray} \frac{{e_{ij}^{n + 1} - e_{ij}^n}}{\tau } = \frac{{e_{ij}^{n + 1} + e_{ij}^n}}{2} \times \frac{{{\Delta _ + }{\Delta _ - }\tilde u_{ij}^n}}{{{h^2}}} + \frac{{u_{ij}^{n + 1} + u_{ij}^n}}{2} \times \frac{{{\Delta _ + }{\Delta _ - }e_{ij}^n}}{{{h^2}}} + o(\tau ) + o({h^2}) \end{eqnarray}$

$\begin{eqnarray} e_{0j}^n = e_{i0}^n = e_{Jj}^n = e_{iJ}^n = 0, (n = 0, 1, \cdots , N) \end{eqnarray}$

$\begin{eqnarray} e_{ij}^0 = 0, (i, j = 0, 1, \cdots , J). \end{eqnarray}$

$(e_{ij}^{n + 1} + e_{ij}^n)\tau h$与(4.1)式作内积, 并对$i, j$从0加到$J$求和得

$\begin{eqnarray} \sum\limits_{i, j = 0}^J {(e_{ij}^{n + 1} - e_{ij}^n) \cdot (e_{ij}^{n + 1} + e_{ij}^n)} h & = & \sum\limits_{i, j = 0}^J {(\frac{{u_{ij}^{n + 1} + u_{ij}^n}}{2} \times \frac{{{\Delta _ + }{\Delta _ - }e_{ij}^n}}{{{h^2}}}) \cdot (e_{ij}^{n + 1} + e_{ij}^n)} \tau h\\&& + \sum\limits_{i, j = 0}^J {o(\tau + {h^2}) \cdot (e_{ij}^{n + 1} + e_{ij}^n)} \tau h, \end{eqnarray}$

$$$||e_{ij}^{n + 1}||_{{L^2}}^2 - ||e_{ij}^n||_{{L^2}}^2\le \tau {c_1}(||e_{ij}^{n + 1}||_{{L^2}}^2 + ||e_{ij}^n||_{{L^2}}^2 + ||\delta e_{ij}^{n + 1}||_{{L^2}}^2 + ||\delta e_{ij}^n||_{{L^2}}^2) + o({\tau ^2} + {h^3}),$$$

$\begin{eqnarray} \sum\limits_{i, j = 0}^J {(e_{ij}^{n + 1} - e_{ij}^n) \cdot \frac{{{\Delta _ + }{\Delta _ - }e_{ij}^n}}{{{h^2}}}} h & = & \sum\limits_{i, j = 0}^J {(\frac{{e_{ij}^{n + 1} + e_{ij}^n}}{2} \times \frac{{{\Delta _ + }{\Delta _ - }\tilde u_{ij}^n}}{{{h^2}}}) \cdot \frac{{{\Delta _ + }{\Delta _ - }e_{ij}^n}}{{{h^2}}}} \tau h\\&& +\sum\limits_{i, j = 0}^J {o(\tau + {h^2}) \cdot \frac{{{\Delta _ + }{\Delta _ - }e_{ij}^n}}{{{h^2}}}} \tau h, \end{eqnarray}$

$\begin{eqnarray} &&||\delta e_{ij}^{n + 1}||_{{L^2}}^2 - ||\delta e_{ij}^n||_{{L^2}}^2{}\\&\le &\tau {c_2}(||e_{ij}^{n + 1}||_{{L^2}}^2 + ||e_{ij}^n||_{{L^2}}^2 + ||\delta e_{ij}^{n + 1}||_{{L^2}}^2 + ||\delta e_{ij}^n||_{{L^2}}^2) + o(\tau + {\tau ^2} + {h^2}), \end{eqnarray}$

$\begin{eqnarray} &&||e_{ij}^{n + 1}||_{{L^2}}^2 + ||\delta e_{ij}^{n + 1}||_{{L^2}}^2 - ||e_{ij}^n||_{{L^2}}^2 - ||\delta e_{ij}^n||_{{L^2}}^2\\ &\le& \tau c(||e_{ij}^{n + 1}||_{{L^2}}^2 + ||e_{ij}^n||_{{L^2}}^2 + ||\delta e_{ij}^{n + 1}||_{{L^2}}^2 + ||\delta e_{ij}^n||_{{L^2}}^2) + o(\tau + {h^2}), \end{eqnarray}$

 时间层 精确解 差分格式解 误差值 所需时间(s) 1 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.017160 2 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.012998 3 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.011783 4 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011667 5 1.00+e00 1.00-6.66134e-16 6.66134e-16 0.011951 6 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011711 7 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.011630 8 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011647 9 1.00+e00 1.00-5.55112e-16 5.55112e-16 0.011787 10 1.00+e00 1.00-5.55112e-16 5.55112e-16 0.011690 11 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011579 12 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011576 13 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.011611 14 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.011501 15 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011609 16 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011973 17 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011524 18 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.012638 19 1.00+e00 1.00-4.44089e-16 4.44089e-16 0.011606 20 1.00+e00 1.00-3.33067e-16 3.33067e-16 0.011690

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