Mathematical Researches
پژوهش های ریاضی
mmr
Basic Sciences
http://mmr.khu.ac.ir
1
admin
2588-2546
2588-2554
10.61186/mmr
fa
jalali
1399
8
1
gregorian
2020
11
1
6
3
online
1
fulltext
fa
حل دستگاه معادلات برگرز دوبعدی با استفاده از طرحهای تفاضلات متناهی نیمه-لاگرانژی
Solving a System of 2D Burger's Equations using Semi-Lagrangian Finite Difference Schemes
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>
<strong>Introduction</strong><br>
Following and generalizing the excellent work of Wang et ‎al. ‎[26], ‎we extract here some new scheme‎<span dir="RTL">s</span>, ‎based on the‎ ‎semi-Lagrangian discretization‎, ‎the modified equation theory‎, ‎and‎ ‎the local one-dimensional (LOD) scheme for computing solutions to a‎ ‎system of two-dimensional (2D) Burgers' equations‎. ‎A careful error‎ ‎analysis is carried out to demonstrate the accuracy of the‎ ‎proposed semi-Lagrangian finite difference methods‎. ‎By conducting‎ ‎numerical simulation to the nonlinear system of 2D Burgers‎’ ‎equations (3.1), ‎we show high accuracy and‎ ‎unconditional stability of the five-point implicit scheme (3.32-3.33)‎. ‎The results of‎ [26] and this paper confirm that the classical modified‎ ‎equation technique can be easily extended to various 1D as well as 2D‎ ‎nonlinear problems‎. ‎Furthermore, a new viewpoint is opened to‎ ‎develop efficient semi-Lagrangian methods‎. ‎Without using suitable‎ ‎interpolants for generating the solution values at the departure‎ ‎points‎, ‎<span dir="RTL">we are not able to</span> apply our method‎. ‎Instead of focusing our‎ ‎concentration on dealing with the effect of various interpolation‎ ‎methods‎, ‎we focus our attention on constructing some‎ ‎explicit and implicit schemes‎. ‎Among various interpolants which‎ ‎can be found in the literature [6], [21], ‎we just‎ ‎exploit the simplest and more applicable interpolants‎, ‎i.e.‎, ‎B-spline and Lagrange interpolants‎. Some semi-Lagrangian schemes are developed using the modified equation‎ ‎approach‎, i.e., ‎a six-point explicit method (which suffers from the‎ ‎limited stability condition)‎, ‎a six-point implicit method (which‎ ‎has unconditional stability but low order truncation error)‎, ‎and a‎ ‎five-point implicit method‎ (3.32-3.33) which has‎ ‎unconditional stability and high order truncation error‎. ‎In each‎ ‎step of this scheme, we must solve two tridiagonal linear systems‎ ‎and therefore its computational complexity is low‎. ‎Furthermore, it‎ ‎can be implemented in parallel‎. ‎As mentioned in [26], ‎this algorithm can be naturally‎ ‎extended to the development of efficient and accurate‎ ‎semi-Lagrangian schemes for many other types of nonlinear‎ ‎time-dependent problems‎, ‎such as the KdV equation and ‎Navier-Stokes equations‎, ‎where advection plays an important role‎. ‎We tried in [9] to apply this approach to the KdV equation but‎ ‎constructing an implicit method which has unconditional stability‎ ‎and high order truncation error needs some considerable symbolic‎ ‎computations for extracting the coefficients of the scheme‎.<br>
<strong>Material and methods</strong><br>
For constructing five-point implicit scheme‎ (3.32-3.33), we need to exploit Lagrange or B-spline interpolation method, ‎‎modified equation approach‎ and ‎local‎<span dir="RTL"> ‎</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 ‎approach, <span dir="RTL">s</span>ome semi-Lagrangian schemes for solving a‎ ‎system of 2D Burgers' equations are developed here which are<span dir="RTL">:</span>
<ul>
<li>A six-point explicit method which is conditionally stable ‎<span dir="RTL"> and its</span> order of truncation error is low,</li>
<li>‎A six-point implicit method which‎ ‎has unconditional stability and its order of truncation error is not high‎,</li>
<li>A five-point implicit method‎ which has‎ ‎unconditional stability, high order truncation error <span dir="RTL">and resonable </span>computational complexity‎.</li>
</ul>
<strong>Conclusion </strong><br>
We encapsulate findings and conclusions of this research as follows:
<ul>
<li>Our‎ ‎proposed scheme is a local one-dimensional scheme which‎ ‎obtained on the basis of the modified equation approach,</li>
<li>Our semi-Lagrangian finite‎ ‎difference scheme is not limited by the‎ ‎Courant‎- ‎Friedrichs-Lewy (CFL) condition and therefore we can‎ ‎apply larger step size for the time variable,</li>
<li>The five-point implicit method‎ proposed is a‎ <span dir="RTL">high order </span>‎unconditionally stable method with<span dir="RTL"> resonable </span>computational costs‎.</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