Konik Marta, modelowanie
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//-->1. Równania ruchu dla u w notacji indeksowej:a). Pochodna przedniammmmmmmmmm mmmum+1−umum+1,kl+um kl−2umum, k+1,l+um,k−1,l−2um1pj+1, kl−pjklKl+1−Klujk ,l+1+ujk , l−1−2ujkljkljklmuj+1,kl-ujklmuj , k+1,l−ujklmujk ,l+1-ujklmjj−1,jkljjjkl+ujkl+vjkl+wjkl−fvjkl=−ρ++Ajk(+)222ΔtΔxΔyΔzΔxΔzΔzΔxΔyΟ(Δt)Ο(Δx)Ο(Δy)Ο(Δz)Ο(Δx)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2um−um−1jkljklΔtb). Pochodna wstecznammmmmmmmmmm−1m−1m−1um−1kl−um−1−2um−1um−1+um−1−2um−11pjkl−pj−1, klKl−Kl−1ujk ,l+1+ujk ,l−1−2ujklm−1ujkl−uj−1,klm−1ujkl−uj , k−1,lm−1ujkl−ujk ,l−1m−1j+1,j−1,kljklj ,k+1,lj , k−1,ljkl+ujkl+vjkl+wjkl−fvjkl=−ρ++Ajk(+)22ΔxΔyΔzΔxΔzΔzΔxΔyΟ(Δt)m+1jklm−1jklΟ(Δx)mj+1, klΟ(Δy)Ο(Δz)Ο(Δx)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2c). Pochodna centralnammmmmmmmm−1m−1m−1u−u−um klum−1kl+um−1−2um−1um−1+um,−1−2um−11pj+1,kl−pj−1,klKl+1−Kl−1ujk ,l+1+ujk , l−1−2ujklm−1uj−1,m−1uj , k+1, l−uj , k−1,lm−1ujk , l+1−ujk , l−1m−1j+1,j−1,kljklj , k+1,lj k−1,ljkl+ujkl+vjkl+wjkl−fvjkl=−ρ++Ajk(+)2222Δt2Δx2Δy2ΔzΔx2ΔzΔzΔxΔy2222222Ο(Δt)Ο(Δx)Ο(Δy)Ο(Δz)Ο(Δx)Ο(Δz)Ο(Δz)Ο(Δx)Ο(Δy)22. Równania ruchu dla v w notacji indeksowej:a). Pochodna przedniavjkl−vjklvj+1, kl+vj−1,kl−2vjklvj , k+1,l+vj , k−1,l−2vjkl1pj+1,kl−pjklKl+1−Klvjk , l+1+vjk , l−1−2vjklmvj+1,kl-vjklmvj , k+1,l-vjklmvjk , l+1-vjklm+ujkl+vjkl+wjkl+fujkl=−ρ++Ajk(+)ΔtΔxΔyΔzΔyΔzΔz2Δx2Δy2Ο(Δt)Ο(Δx)Ο(Δy)Ο(Δz)Ο(Δy)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2b). Pochodna wstecznammmmmmmmmmm−1m−1m−1vm−vm−1vm−1−vm−1−2vm−1vm−1+vm−1−2vm−11pjkl−pj−1, klKl−Kl−1vjk , l+1+vjk ,l−1−2vjkljkljklm−1vjkl−vj−1, klm−1vjkl−vj ,k−1,lm−1vjkl−vjk ,l−1m−1jkljkl+ujkl+vjkl+wjkl+fujkl=−ρ++Ajk(j+1,klj−1,kl+j ,k+1,lj ,k−1,l)22ΔtΔxΔyΔzΔyΔzΔzΔxΔym+1mmmmmmmmmmmmmmmmmmmmΟ(Δt)Ο(Δx)Ο(Δy)Ο(Δz)Ο(Δy)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2c). Pochodna centralnammmmmmmmmm−1m−1m−1m−1m−1m−1m−1m−1m−1v−v−vj−1,klvj+1,kl+vj−1,kl−2vjklvj , k+1,l+vj , k−1,l−2vjkl1pj+1,kl−pj−1,klKl+1−Kl−1vjk , l+1+vjk ,l−1−2vjklm−1vm−1vj , k+1,l−vj ,k−1,lm−1vjk ,l+1−vjk ,l−1+ujkl+vjkl+wjkl+fu=−ρ++Ajk(+)2Δt2Δx2Δy2ΔzΔy2ΔzΔz2Δx2Δy2Ο(Δt)2Ο(Δx)2Ο(Δy)2Ο(Δz)2Ο(Δy)Ο(Δz)2Ο(Δz)2Ο(Δx)2Ο(Δy)2m+1jklm−1jklmj+1,kl3. Równania ruchu dla transportu T:a). Pochodna przedniaT−TjklmTj+1,kl-T+ujklΔtΔxΟ(Δt)Ο(Δx)m−1jklm+1jklmmmjkl+vmjklTj , k+1,l-TjklmTjk ,l+1-T+wjklΔyΔzΟ(Δy)Ο(Δz)mmmmjklKl+1−KlTjk , l+1+Tjk ,l−1−2TjklT=+Ajk(ΔzΔz2Ο(Δz)Ο(Δz)2mmmmmmj+1, kl+Tj−1,kl−2TjklT+Δx2Ο(Δx)2mmmj , k+1,l+Tj , k−1,l−2Tjkl)+ΘT2ΔyΟ(Δy)2mmT−TΔtΟ(Δt)mjklb). Pochodna wstecznammmmmmKm−Km−1Tm−1+1+Tm−1−2Tm−1Tm−1−Tm−1 kl−2Tm−1Tm−1+Tm−1 l−2Tm−1m−1Tjkl−Tj−1,klm−1Tjkl−Tj ,k−1,lm−1Tjkl−Tjk , l−1lljk ,ljk , l−1jklj+1,klj−1,jklj ,k+1,lj ,k−1,jkl+ujkl+vjkl+wjkl=+Ajk(+)+ΘT22ΔxΔyΔzΔzΔzΔxΔyΟ(Δx)Ο(Δy)Ο(Δz)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2c). Pochodna centralnammmmmmTm+1−Tm−1Km+1−KmTm−1+1+Tm−1−1−2Tm−1Tm−1kl+Tm−1−2Tm−1Tm,−1+Tm−1−2Tm−1jkljklm−1Tj+1, kl−Tj−1, klm−1Tj ,k+1,l−Tj ,k−1,lm−1Tjk ,l+1−Tjk ,l−1ll−1jk , ljk ,ljklj+1,j−1,kljklj k+1,lj , k−1,ljkl+ujkl+vjkl+wjkl=+Ajk(+)+ΘT2Δt2Δx2Δy2Δz2ΔzΔz2Δx2Δy2Ο(Δt)2Ο(Δx)2Ο(Δy)2Ο(Δz)2Ο(Δz)2Ο(Δz)2Ο(Δx)2Ο(Δy)24. Równanie ruchu dla transportu S:a). Pochodna przedniammmmmmSm+1−SmKm+1−KlmSm ,l+1+Sm ,l−1−2SmSm+1,kl+Sm kl−2SmSm,k+1,l+Sm, k−1,l−2SmjkljklmSj+1,kl-SjklmSj , k+1,l-SjklmSjk ,l+1-Sjklljkjkjkljj−1,jkljjkl+ujkl+vjkl+wjkl=+Ajk(+j)222ΔtΔxΔyΔzΔzΔzΔxΔyΟ(Δt)Ο(Δx)b). Pochodna wstecznaΟ(Δy)Ο(Δz)Ο(Δz)Ο(Δz)2Ο(Δx)2Ο(Δy)2Sjkl−SjklKl−Kl−1Sjk ,l+1+Sjk , l−1−2Sjklm−1Sjkl−Sj−1,klm−1Sjkl−Sj ,k−1,lm−1Sjkl−Sjk , l−1+ujkl+vjkl+wjkl=ΔtΔxΔyΔzΔzΔzΟ(Δt)Ο(Δx)Ο(Δy)Ο(Δz)Ο(Δz)Ο(Δz)2mm−1mmmmmmmmm−1m−1m−1Sj+1,kl−Sj−1,kl−2Sjkl+Ajk(Δx2Ο(Δx)2m−1m−1m−1Sj , k+1,l+Sj ,k−1,l−2Sjkl+)Δy2Ο(Δy)2m−1m−1m−1S−S2ΔtΟ(Δt)2m+1jklm−1jklc). Pochodna centralnammmmmmKm+1−KmSm−11+Sm−1−1−2Sm−1Sm−1+Sm−1 kl−2Sm−1Sm−1 l+Sm,−1−2Sm−1m−1Sj+1, kl−Sj−1, klm−1Sj , k+1,l−Sj ,k−1,lm−1Sjk , l+1−Sjk , l−1ll−1jk , l+jk ,ljklj+1,klj−1,jklj , k+1,j k−1,ljkl+ujkl+vjkl+wjkl=+Ajk(+)2222Δx2Δy2Δz2ΔzΔzΔxΔyΟ(Δx)2Ο(Δy)2Ο(Δz)2Ο(Δz)2Ο(Δz)2Ο(Δx)2Ο(Δy)2
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