A numerical solution for advection-diffusion equation based on a semi-Lagrangian scheme

Abstract

This paper describes a numerical solution for the advection-diffusion equation.  The proposed method is based on the operator splitting method which helps to obtain accurate solutions.  That is, instead of sum, the operators are considered separately for the physical compatibility.  In the process, method of characteristics combined with cubic spline interpolation and Saulyev method are used in sub-operators, respectively.  After guaranteeing the convergence of the method the efficiency is also tested on one-dimensional advection-diffusion problem for a wide range of Courant numbers which plays a crucial role on the convergence of the solution.  The obtained results are compared with the analytical solution of the problem and other solutions which are available in the literature.  It is revealed that the proposed method produces good approach not only for small Caurant numbers but also big ones even though it is explicit method.

Keywords

• Advection-diffusion equation,method of characteristics,Saulyev scheme,numerical solution

Yarı-Lagrangian bir şema yaklaşımına dayalı adveksiyon-difüzyon denkleminin bir sayısal çözümü

Abstract

Bu çalışmada, adveksiyon-difüzyon denklemi için sayısal bir çözüm tanıtılmaktadır. Önerilen yöntem, doğru çözümler elde edilmesine yardımcı olan operatör ayırma metoduna dayanmaktadır.  Yani, toplam yerine, operatörler fiziksel uyumluluk için ayrı olarak ele alınmaktadır.  Bu süreçte, alt operatörler için karakteristikler yöntemi ile bir araya getirilmiş kübik spline interpolasyonu ve Saulyev metodu sırası ile kullanılmıştır.  Yöntemin yakınsamasını garanti altına aldıktan sonra, verimlilik de çözümün yakınsaması üzerinde önemli bir rol oynayan farklı Courant sayıları için tek boyutlu adveksiyon-difüzyon problemi üzerinde test edilmiştir.  Elde edilen sonuçlar, problemin analitik çözümü ve literatürde mevcut olan diğer çözümlerle karşılaştırılmıştır.  Önerilen yöntemin, açık bir yöntem olmasına rağmen sadece küçük Caurant sayıları için değil, büyük olanlar için de iyi bir yaklaşım oluşturduğu ortaya çıkmıştır.

Keywords

• Adveksiyon-difüzyon denklemi,karakteristikler metodu,Saulyev şeması

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