Equação de Evolução e método do volume-finito.

Apresentação em tema: "Equação de Evolução e método do volume-finito."— Transcrição da apresentação:

1 Equação de Evolução e método do volume-finito.
4ª aula Equação de Evolução e método do volume-finito.

2 Objectivo da Aula Fazer a “mise à zero” como dizem os Franceses.
Mostrar que o princípio de conservação pode ser descrito por : Palavras (descrição do conceito) Na forma de uma equação integral, Na forma de uma equação diferencial (que nalguns casos pode ter solução analítica), Na forma de uma equação algébrica (usada pelos modelos matemáticos). Apresentar métodos para o cálculo da advecção. Discretização temporal, estabilidade e difusão numérica.

3 Taxa de acumulação M t1 t1+Δt ΔM Δt t

4 Taxa de acumulação

5 Fluxos no caso 1D No caso 1D (e.g. um rio) as propriedades só variam ao longo de uma direção. Poderemos por isso representar o Sistema por uma fila de volumes de control.

6 Adicionando

7 Fazendo o volume tender para zero
Admitindo que o volume é um paralelepípedo que não varia no tempo: Sabendo que

8 Equação diferencial Num ponto, no caso 1D o princípio de conservação estabelece que: No caso 3D:

9 A forma algébrica requer hipóteses adicionais
A forma algébrica requer hipóteses adicionais. Método “upwind” com Q>0

10 Condição de estabilidade do método upwind

11 Average values on faces => Central Differences

12 Diferenças Centrais Explícitas
Condição de estabilidade:

13 Understanding the Central Differences
Why are CD instable without diffusion? Resp: They violate the transportive property of advection. The computing point learns about the downstream property value through advection, which is physically impossible. Why can be stable with diffusion Resp: Because diffusion transports information in any direction. If the diffusive transport is stronger than advective, the process becomes physically correct.

14 More Questions Can explicit central differrences be used on advection is dominant? Resp: No. In that case difusion transports upstream much less than advection transport downstream (Grid Reynolds number is large). If diffusion is dominant is better to use centra differences or upwind? Central differences are better they have second order accuracy and introduce less numerical diffusion. What about an implicit algorithm? Would it be stable without diffusion? Resp: Yes. In implicit algorithms fluxes are computed using the new concentrations. If Advection would generate negative concentrations the leaving flux would become positive. Thos means that it is impossible to generate negative concentrations. Even in upwind? Resp: In upwind case the concentration can become negative only if we remove from a volume more than its content. But since what is leaving the volume is computed at the end of the time step negative values can not be generated.

15 Other methods for advection
Upwind: Assumes that the concentration at the face is equal to the upstream concentration. Central Differences: Assume the average between both sides. What if a 2nd order polynom was considered (using 3 points)? One would get the QUICK: (Quadratic Upstream Interpolation for Convective Kinematics): It has 3rd order accuracy. It has increased stability problems next to boundaries. What is the best method?

16 Algorithm for implicit methods

17 The Semi-implicit method (Crank – Nicholson)

18 Summary The finite control volume helps us to think about fluxes. That is usually good.. Explicit methods get instable when the amount removed from a volume is higher than the volume contained inside. Central differences assume linear concentration evolution between adjacent finite volumes. They cannot respect the transportive property of advection. The QUICK method assume a quadratic evolution between adjacent centres. Requires three points and consequently cannot be used next to boundaries. It does not respect completely the transportive property and can generate instabilities. It is better if combined with upwind. The finite volume method puts into evidence the advantage of combining methods for advection.