Comparison of Finite Difference Solution Methods for Reaction Diffusion System in Two Dimensions

Abstract

In this paper, we study three types of finite difference methods, to find the numerical solution of reaction difference systems of PDEs in two dimensions. These methods are ADE, ADI and Hopscotch, where Gray-Scott model in two dimensions has been considered. Our numerical results show that the ADI method produces more accurate and stable solution than ADE method and Hopscotch method is the best because does not involve any tridiagonal matrix. Also we studied the consistency, stability and convergence of the above methods. Keywords: Gray-Scott model in two dimensions, finite difference methods, Alternating direction explicit method, Alternating direction implicit method, Hopscotch method. ‫بعدين‬ ‫في‬ ‫التفاعلي‬ ‫االنتشار‬ ‫لنظام‬ ‫المنتهية‬ ‫الفروقات‬ ‫حلول‬ ‫ائق‬ ‫طر‬ ‫مقارنة‬‫البياتي‬ ‫يونس‬ ‫عباس‬ 1 ‫مناع‬ ‫عبدهللا‬ ‫سعد‬ 2 ‫الرو‬ ‫امين‬ ‫محمد‬ ‫عبدالغفور‬ ‫ژ‬ ‫بياني‬ 3 1,3 ‫اق‬ ‫الموصل/العر‬ ‫ياضيات/جامعة‬ ‫الر‬ ‫و‬ ‫الحاسوب‬ ‫علوم‬ ‫كلية‬ 2 ‫اق‬ ‫اخو/العر‬ ‫ز‬ ‫العلوم/جامعة‬ ‫كلية‬ ‫البحث:‬ ‫استالم‬ ‫تاريخ‬ 23 / 4 / 2007 ‫البحث:‬ ‫قبول‬ ‫تاريخ‬ 15 / 8 / 200 ‫الملخص‬ ‫ف‬ ‫ي‬ ‫البحث‬ ‫هذا‬ ‫المنتهية‬ ‫الفروقات‬ ‫طرق‬ ‫من‬ ‫اع‬ ‫أنو‬ ‫ثالث‬ ‫ندرس‬ ، ‫وبالتفصيل‬ ، ‫ان‬ ‫م‬ ‫اام‬ ‫لنظ‬ ‫العددي‬ ‫الحل‬ ‫إليجاد‬ ‫ا‬ ‫ا‬ ‫ه‬ ‫ن،‬ ‫اد‬ ‫ا‬ ‫ع‬ ‫اي‬ ‫ا‬ ‫ف‬ ‫ااف‬ ‫ا‬ ‫الم‬ ‫اي‬ ‫ا‬ ‫و‬ ‫ال‬ ‫اوع‬ ‫ا‬ ‫ن‬ ‫ان‬ ‫ا‬ ‫م‬ ‫اة‬ ‫ا‬ ‫النوي‬ ‫ار‬ ‫ا‬ ‫خي‬ ‫اة‬ ‫ا‬ ‫غي‬ ‫الج‬ ‫الية‬ ‫ا‬ ‫التفاض‬ ‫ت‬ ‫ااد‬ ‫ا‬ ‫المع‬ ‫ذ‬ ‫اات‬ ‫ا‬ ‫المتجه‬ ‫اة‬ ‫ا‬ ‫ي‬ ‫طر‬ ‫اي‬ ‫ا‬ ‫ه‬ ‫ارق‬ ‫ا‬ ‫الو‬ ‫ه‬ ‫يحة‬ ‫ار‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫الص‬ ‫اة‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫المتعاقب‬ ADE ‫امنية‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ال‬ ‫اة‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫المتعاقب‬ ‫اات‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫المتجه‬ ‫اة‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ي‬ ‫وطر‬ ) (ADI) ‫و‬ ‫اة‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ي‬ ‫طر‬ Hopscotch ‫او‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫ا‬ ‫لنم‬ ) Gray-Scott ‫اء‬ ‫ف‬ ‫في‬ ‫منية‬ ‫ال‬ ‫المتعاقبة‬ ‫المتجهات‬ ‫ة‬ ‫ي‬ ‫طر‬ ‫أن‬ ‫العددية‬ ‫النتاغج‬ ‫من‬ ‫استنتجنا‬ ‫و‬ ‫ن.‬ ‫عد‬ ‫ات‬ ADI ) ‫يحة‬ ‫ار‬ ‫ا‬ ‫الص‬ ‫اة‬ ‫ا‬ ‫المتعاقب‬ ‫اات‬ ‫ا‬ ‫المتجه‬ ‫اة‬ ‫ا‬ ‫ي‬ ‫طر‬ ‫ان‬ ‫ا‬ ‫م‬ ‫ال‬ ‫ا‬ ‫أف‬ ‫اي‬ ‫ا‬ ‫ه‬ ADE ‫اة‬ ‫ا‬ ‫ي‬ ‫طر‬ ‫رن‬ ‫اة‬ ‫ا‬ ‫النتيج‬ ‫ان‬ ‫ا‬ ‫ولك‬ ) Hopscotch ‫ال‬ ‫ا‬ ‫أف‬ ‫اي‬ ‫ا‬ ‫ه‬ ‫اة‬ ‫ي‬ ‫طر‬ ‫من‬ ADI ‫ا‬ ‫ي‬ ‫ن‬ ‫ا‬ ‫خ‬ ‫از‬ ‫رل‬ ‫اا‬ ‫ححت‬ ‫اا‬ ‫ألنه‬ ‫اار‬ ‫األقو‬ ‫اة‬ ‫الاالثي‬ ‫افوفات‬ ‫المص‬ ‫ايمة‬ ‫ص‬ ‫اوي‬ ‫ححت‬ ‫اا‬ ‫أنه‬ ‫از‬ ‫رل‬ ‫اافة‬ ‫اإلض‬ ‫ر‬ Tridiagonal ‫الاالثة.‬ ‫الورق‬ ‫من‬ ‫لكل‬ ‫ارب‬ ‫وح‬ ‫ية‬ ‫ار‬ ‫ر‬ ‫است‬ ‫و‬ ‫وحية‬ ‫ث‬ ‫درسنا‬ ‫ما‬ . ‫المفتاحية:‬ ‫الكلمات‬ ‫گراي‬ ‫أنموذج‬ -‫طريقة‬ , ‫الصريحة‬ ‫المتعاقبة‬ ‫المتجهات‬ ‫طريقة‬ ‫المنتهية,‬ ‫الفروقات‬ ‫طرائق‬ ‫سكوت,‬ ‫ال‬ ‫المتعاقبة‬ ‫المتجهات‬ ‫هبسكوج.‬ ‫طريقة‬ , ‫ضمنية‬

Stabil, explicit és implicit numerikus algoritmusok tesztelése vezetéses hővezetési feladatokra

“…ahol az i index a magasabb dimenziójú, inhomogén rendszer celláit "címkézi". i I -n az i -edik cellával szomszédos cellák indexeinek a halmazát értjük, tehát az iménti (20) egyenletben szereplő összegzés a megfelelő szomszédos cellákra történik. A (20) kifejezés megadható egy tömörebb mátrixegyenlet formájában is:…”

Comparison of the Performance of New and Traditional Numerical Methods for Long-Term Simulations of Heat Transfer in Walls with Thermal Bridges

“…In light of the aforementioned data, it is reasonable to assume that explicit algorithms, particularly if they have improved stability qualities (see, for example, [25][26][27][28][29][30][31][32]), will have a growing comparative advantage over time. We started working on new explicit schemes a couple of years ago for determining heat conduction in any number of spatial dimensions.…”

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

“…One can observe that the trend toward increasing parallelism in high-performance computing is reinforced, since unfortunately the CPU clock frequencies nowadays increase much slower than a few decades ago [17,18]. That is one of the reasons why we believe that the importance of easily parallelizable explicit and unconditionally stable methods is going to increase, even if currently not too many scholars work with them (see [19][20][21][22][23][24][25][26] for examples).…”