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# Day21-slides - Atms 502 Numerical Fluid Dynamics Thu Nov 2...

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1 Atms 502 Numerical Fluid Dynamics Thu., Nov. 2, 2006 Monotonicity; van Leer (1977) methods Conservative, monotonic schemes References: Wilhelmson, Wicker Notes Droegemeier notes Durran chapters. 3, 5 11/6/06 Atms 502 - Fall 2006 - Jewett 3 Monotonicity A monotonic scheme is one which: 1 Given an initial distribution that is monotonic before advection … Produces a distribution that is monotonic after advection. Durran makes an important distinction: A monotone scheme guarantees that 2 : In a monotonicity-preserving scheme: 1 Droegemeier if a j n " b j n for all j, at time n, then a j n + 1 " b j n + 1 will hold for all j and n. 2 wilhelmson if a j 0 " a j + 1 0 for all j, at time 0, then a j n " a j + 1 n will hold for all j and n. 11/6/06 Atms 502 - Fall 2006 - Jewett 4 van Leer grid zone averages piecewise constant van Leer: Upstream scheme Familiar… grid point values q n + 1 = q n " c # t # x q i " q i " 1 ( ) = q n " cq i " cq i " 1 [ ] # t # x q 1/ 2 = q 1/ 2 " u q 1/ 2 " u q " 1/ 2 [ ] # t # x

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2 11/6/06 Atms 502 - Fall 2006 - Jewett 5 Monotonicity When we refer to monotonicity we will mean
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Day21-slides - Atms 502 Numerical Fluid Dynamics Thu Nov 2...

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