2ª aula Evolution Equation. The Finite Volume Method.

Objective of the lecture 1.The Students “mise à zero” as francophone say. 2.To show that the conservation principle can be written on: – Words (is a concept) – Integral equation form, – Differential form (can have analytical solution), – Algebraic form (used by mathematical models). 3.To recall methods to solve advection. Temporal discretisation, stability and numerical diffusion. 4.To tell about the need for initial and boundary conditions and about the difficulties to get them.

Fluxes: – Advective: (Why the sign “-”)? – Diffusive: The Integral Equation Accumulation Rate:

Differential Evolution Equation The rate of accumulation is “minus” the divergence of the fluxes + (Souces-Sinks) Or:

Finite Volume The Finite (Control) Volume : 1) isolates a portion of the space, 2) systematises budgets’ computation across its faces, 3) Computes the rate of accumulation, 4) Permits the computation of a property rate of change.

Computational grid

Assuming uniform concentration inside the volume one gets

Finite Volume in a 1D case In 1D case properties can change along one direction only.

Summing up

Shrinking the volume to zero That is the 1D advection-diffusion equation for one property. In a 3D case, for a generic property “k” one would get: That represents the conservation principle in one point And assuming that the volume is a parallelepiped that doesn’t change in time: Knowing that:

Algebraic form Requires hypothesis. The upwind formulation with Q>0

Hypothesis impose stability conditions Stability condition:

Average values on faces => Central Differences

Explicit Central Differences Stability conditions:

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.

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.

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 3 rd order accuracy. It has increased stability problems next to boundaries. What is the best method?

Algorithm for implicit methods

The Semi-implicit method (Crank – Nicholson)

Initial and Boundary Conditions Initial conditions are less important in dissipative systems (high sinks). Boundary conditions are less important when Sources and/or sinks are important? CiCi C i-1 C i+1

Boundary conditions Diffusion: – Requires the knowledge of concentrations outside the domain. If not known zero gradient is usually the best option Advection – When the flow enters the domain carries information from outside that must be known.

How to know the external boundary condition? Gradients can be considered null if sources and sinks are important. Otherwise nested models are required!

¼° Downscaling 23

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. Initial and boundary conditions choice determine the quality of the results.