fa حل دستگاه معادلات برگرز دوبعدی با استفاده از طرح‌های تفاضلات متناهی نیمه-لاگرانژی جبر alg مقاله استخراج شده از پایان نامه Research Paper <span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">در این مقاله قصد داریم طرح‌های تفاضلات متناهی نیمه-لاگرانژی را برای دستگاه معادلات برگرز دوبعدی تعمیم دهیم. طرح پیشنهادی به شرط کورانت-فردریش-لوی (</span></span><span dir="LTR"><span style="font-size:10.0pt;">CFL</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">) محدود نیست و بنابراین می‌توان اندازه گام‌های زمانی بزرگی انتخاب کرد. طرح پیشنهادی قابلیت موازی‌سازی خوبی دارد و در اصل یک طرح یک-بعدی موضعی </span></span><span dir="LTR"><span style="font-size:10.0pt;">(LOD)</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> است که بر اساس راه‌کار معادله تغییر یافته به‌دست آمده است و برای حل دستگاه معادلات برگرز به‌کار می‌رود. یک ویژگی خوب روش مطرح‌ شده آن است که در هر تکرار زمانی کافی است دو دستگاه خطی سه‌قطری حل شود و از این نظر حجم محاسباتی روش پائین است.</span></span><br> &nbsp; <strong>Introduction</strong><br> Following and generalizing the excellent work of Wang et &lrm;al. &lrm;[26], &lrm;we extract here some new scheme&lrm;<span dir="RTL">s</span>, &lrm;based on the&lrm; &lrm;semi-Lagrangian discretization&lrm;, &lrm;the modified equation theory&lrm;, &lrm;and&lrm; &lrm;the local one-dimensional (LOD) scheme for computing solutions to a&lrm; &lrm;system of two-dimensional (2D) Burgers&#39; equations&lrm;. &lrm;A careful error&lrm; &lrm;analysis is carried out to demonstrate the accuracy of the&lrm; &lrm;proposed semi-Lagrangian finite difference methods&lrm;. &lrm;By conducting&lrm; &lrm;numerical simulation to the nonlinear system of 2D Burgers&lrm;&rsquo; &lrm;equations (3.1), &lrm;we show high accuracy and&lrm; &lrm;unconditional stability of the five-point implicit scheme (3.32-3.33)&lrm;. &lrm;The results of&lrm; [26] and this paper confirm that the classical modified&lrm; &lrm;equation technique can be easily extended to various 1D as well as 2D&lrm; &lrm;nonlinear problems&lrm;. &lrm;Furthermore, a new viewpoint is opened to&lrm; &lrm;develop efficient semi-Lagrangian methods&lrm;. &lrm;Without using suitable&lrm; &lrm;interpolants for generating the solution values at the departure&lrm; &lrm;points&lrm;, &lrm;<span dir="RTL">we are not able to</span> apply our method&lrm;. &lrm;Instead of focusing our&lrm; &lrm;concentration on dealing with the effect of various interpolation&lrm; &lrm;methods&lrm;, &lrm;we focus our attention on constructing some&lrm; &lrm;explicit and implicit schemes&lrm;. &lrm;Among various interpolants which&lrm; &lrm;can be found in the literature [6], [21], &lrm;we just&lrm; &lrm;exploit the simplest and more applicable interpolants&lrm;, &lrm;i.e.&lrm;, &lrm;B-spline and Lagrange interpolants&lrm;. Some semi-Lagrangian schemes are developed using the modified equation&lrm; &lrm;approach&lrm;, i.e., &lrm;a six-point explicit method (which suffers from the&lrm; &lrm;limited stability condition)&lrm;, &lrm;a six-point implicit method (which&lrm; &lrm;has unconditional stability but low order truncation error)&lrm;, &lrm;and a&lrm; &lrm;five-point implicit method&lrm; (3.32-3.33) which has&lrm; &lrm;unconditional stability and high order truncation error&lrm;. &lrm;In each&lrm; &lrm;step of this scheme, we must solve two tridiagonal linear systems&lrm; &lrm;and therefore its computational complexity is low&lrm;. &lrm;Furthermore, it&lrm; &lrm;can be implemented in parallel&lrm;. &lrm;As mentioned in [26], &lrm;this algorithm can be naturally&lrm; &lrm;extended to the development of efficient and accurate&lrm; &lrm;semi-Lagrangian schemes for many other types of nonlinear&lrm; &lrm;time-dependent problems&lrm;, &lrm;such as the KdV equation and &lrm;Navier-Stokes equations&lrm;, &lrm;where advection plays an important role&lrm;. &lrm;We tried in&nbsp; [9] to apply this approach to the KdV equation but&lrm; &lrm;constructing an implicit method which has unconditional stability&lrm; &lrm;and high order truncation error needs some considerable symbolic&lrm; &lrm;computations for extracting the coefficients of the scheme&lrm;.<br> <strong>Material and methods</strong><br> For constructing five-point implicit scheme&lrm; (3.32-3.33), we need to exploit Lagrange or B-spline interpolation method, &lrm;&lrm;modified equation approach&lrm; and &lrm;local&lrm;<span dir="RTL"> &lrm;</span>one-dimensional technique. The five-point implicit scheme is unconditional stable, has satisfactory order of convergence and its computational costs is low.<br> <strong>Results and discussion</strong><br> Using the modified equation&lrm; &lrm;approach, <span dir="RTL">s</span>ome semi-Lagrangian schemes for solving a&lrm; &lrm;system of 2D Burgers&#39; equations are developed here which are<span dir="RTL">:</span> <ul> <li>A six-point explicit method which is conditionally stable &lrm;<span dir="RTL"> and its</span> order of truncation error is low,</li> <li>&lrm;A six-point implicit method which&lrm; &lrm;has unconditional stability and its order of truncation error is not high&lrm;,</li> <li>A five-point implicit method&lrm; which has&lrm; &lrm;unconditional stability, high order truncation error &nbsp;<span dir="RTL">and resonable </span>computational complexity&lrm;.</li> </ul> <strong>Conclusion </strong><br> We encapsulate findings and conclusions of this research as follows: <ul> <li>Our&lrm; &lrm;proposed scheme is a local one-dimensional scheme which&lrm; &lrm;obtained on the basis of the modified equation approach,</li> <li>Our semi-Lagrangian finite&lrm; &lrm;difference scheme is not limited by the&lrm; &lrm;Courant&lrm;- &lrm;Friedrichs-Lewy (CFL) condition and therefore we can&lrm; &lrm;apply larger step size for the time variable,</li> <li>The five-point implicit method&lrm; proposed is a&lrm; <span dir="RTL">high order </span>&lrm;unconditionally stable method with<span dir="RTL"> resonable </span>computational costs&lrm;.</li> </ul> دستگاه معادلات برگرز, طرح تفاضلات متناهی نیمه-لاگرانژی, راه‌کار معادله تغییریافته, طرح یک-بعدی موضعی. System of 2D Burgers' equations‎, ‎Semi-Lagrangian‎ ‎finite difference scheme‎, ‎Modified equation approach‎, Local‎ ‎one-dimensional approach‎‎. 449 464 http://mmr.khu.ac.ir/browse.php?a_code=A-10-412-1&slc_lang=fa&sid=1 Reza Mokhtari رضا مختاری mokhtari@cc.iut.ac.ir 10031947532846003971 10031947532846003971 Yes Isfahan University of Technology دانشگاه صنعتی اصفهان، دانشکدۀ علوم ریاضی Elham Feizollahi الهام فیض‌اللهی e.feizollahi@math.iut.ac.ir 10031947532846003972 10031947532846003972 No Isfahan University of Technology دانشگاه صنعتی اصفهان، دانشکدۀ علوم ریاضی