Numerical Solutions of Advection-Diffusion Equation via Quartic B-Spline Least Squares Method

Yıl 2023, Cilt: 4 Sayı: 3, 9 - 14, 27.12.2023

Buket Ay , Bülent Saka , Ozlem Ersoy Hepson , İdris Dağ

https://doi.org/10.53608/estudambilisim.1310048

Öz

Numerical solutions are obtained for the time dependent one dimensional Advection-Diffusion equation (ADE) using the least squares method (LSM). After the time-integration of the ADE by use of the Crank-Nicolson technique, the combination of the quartic B-splines as an trial function is substituted in the time discretized ADE and for the LSM procedure, a weight function is obtained by taking derivative of the integrand of integral of squared ADE with respect to time parameters of the trial functions. Thus solution of obtained recursive system of the unknown parameters provides solutions of the ADE at discrete times. Achievement is displayed by studying three test problems.

Anahtar Kelimeler

Advection-Diffusion Denklemi, B-splines, Least Squares Method

Adveksiyon Difüzyon Denkleminin Kuartik B-Spline En Küçük Kareler Metodu ile Sayısal Çözümü

Öz

En küçük Kareler metodu (LSM) kullanılarak zamana bağlı bir boyutlu Advection-Diffusion denklemi (ADE) için yaklaşık çözümler elde edilmiştir. Crank-Nicolson yöntemi ile ADE’in zaman ayrıştırması yapıldıktan sonra, Kuadratik B-spline fonksiyonların kombinasyonundan oluşan deneme fonksiyonu zaman ayrışmış ADE’nin bilinmeyen fonksiyonuna yerleştirilmiştir. Böylece elde edilen cebirsel ifadenin karesi alınmış integralinin integrantının deneme fonksiyonun da tanımlı zaman parametresine göre türevi alınarak ağırlık fonksiyonu belirlenmiştir. Bulunan iterative sistemin çözümlerinden ADE’nin ayrık zamanlardaki çözümlerini elde edilir. Metodun başarısı üç test problem çalışılarak gösterilmiştir.

Anahtar Kelimeler

Adveksiyon-Difuzyon Denklemi, B-splines, En Küçük Kareler Yöntemi

Kaynakça

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