حل دستگاه معادلات برگرز دوبعدی با استفاده از طرح‌های تفاضلات متناهی نیمه-لاگرانژی

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 (cfl) محدود نیست و بنابراین می‌توان اندازه گام‌های زمانی بزرگی انتخاب کرد. طرح پیشنهادی قابلیت موازی‌سازی خوبی دارد و در اصل یک طرح یکبعدی موضعی (lod) 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.

Solving a System of 2D Burger's Equations using Semi-Lagrangian Finite Difference Schemes

IntroductionFollowing and generalizing the excellent work of Wang et al. [26], we extract here some new scheme s, based on the semiLagrangian discretization , the modified equation theory , and the local onedimensional (LOD) scheme for computing solutions to a system of twodimensional (2D) Burgerschr('39') equations A careful error analysis is carried out to demonstrate the accuracy of the proposed semiLagrangian finite difference methods By conducting numerical simulation to the nonlinear system of 2D Burgers rsquo; equations (3.1), we show high accuracy and unconditional stability of the fivepoint implicit scheme (3.323.33) 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 semiLagrangian methods Without using suitable interpolants for generating the solution values at the departure points , we are not able to 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. , Bspline and Lagrange interpolants Some semiLagrangian schemes are developed using the modified equation approach , i.e., a sixpoint explicit method (which suffers from the limited stability condition) , a sixpoint implicit method (which has unconditional stability but low order truncation error) , and a fivepoint implicit method (3.323.33) 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 semiLagrangian schemes for many other types of nonlinear timedependent problems , such as the KdV equation and NavierStokes 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 .Material and methodsFor constructing fivepoint implicit scheme (3.323.33), we need to exploit Lagrange or Bspline interpolation method, modified equation approach and local onedimensional technique. The fivepoint implicit scheme is unconditional stable, has satisfactory order of convergence and its computational costs is low.Results and discussionUsing the modified equation approach, some semiLagrangian schemes for solving a system of 2D Burgerschr('39') equations are developed here which are:A sixpoint explicit method which is conditionally stable and its order of truncation error is low, A sixpoint implicit method which has unconditional stability and its order of truncation error is not high ,A fivepoint implicit method which has unconditional stability, high order truncation error and resonable computational complexity .Conclusion We encapsulate findings and conclusions of this research as follows:Our proposed scheme is a local onedimensional scheme which obtained on the basis of the modified equation approach,Our semiLagrangian finite difference scheme is not limited by the Courant FriedrichsLewy (CFL) condition and therefore we can apply larger step size for the time variable,The fivepoint implicit method proposed is a high order unconditionally stable method with resonable computational costs .