• For Multi-variable block solution like the compressible Navier-Stokes system above, the same trick is used: the cell variable consists of (ρ, ρu, ρE) and the coupling can be coupled by a 5 × 5 matrix coefficient
• Important disadvantages of a block coupled system are
– Large linear system: several variables are handled together
– Different kinds of physics can be present, e.g. the transport-dominated momentum equation and elliptic pressure equation. At matrix level, it is impossible to separate them, which makes the system more difficult to solve
Nature of Coupling
• Block matrix represents complete coupling for a block variable
• We can examine cases of partial coupling by looking at degenerate forms of the coefficients. This will reveal special cases of coupling where alternatives to a fully coupled solution approach may be considered
8.4 Matrix Structure for Coupled Algorithms
Matrix Connectivity and Mesh Structure
• Irrespective of the level of coupling, the FVM dictates that a cell value will depend only on the values in surrounding cells
P
W E
N
S
• We still have freedom to organise the matrix by ordering entries for various components of φ. Also, the matrix connectivity pattern may be changed by reordering the computational points
• Example: block-coupled vector equation (ux, uy, Uz)
– Per-variable organisation: first ux for all cells, followed by uy and uz. Ordering of each sub-list matches the cell ordering.
aP = of computational points and contain the coupling within the single component. All matrix coefficients are scalars. Off-diagonal block represent variable-to-variable coupling.
– Per-cell organisation: (ux, uy, Uz) for each cell. A single numbering space for all cells, but each individual coefficient is more complex:
contains complete coupling
• Both choices have advantages and choice depends on software infrastructure and matrix assembly methods. In order to illustrate the nature of coupling, we shall choose per-cell organisation
Coupling Coefficient
• Consider a linear dependence between two vectors m and n. We can write a general form as
m = A b (8.30)
We shall evaluate the shape of A for various levels of coupling. We shall think of A as a matrix coefficient in the block matrix. The diagonal matrix entry is termed AP and the off-diagonal as AN. Matrix connectivity is dictated by the mesh structure
• Component-wise coupling describes the case where mx depends only on nx, my on ny and mz on nz
1. Scalar component-wise coupling 2. Vector component-wise coupling 3. Full (block) coupling
• Explicit methods do not feature here because it is not necessary to express them in terms of matrix coefficients
• For reference, the linear equation for each cells featuring in the matrix reads APmP +X
N
ANmN = R (8.31)
8.4 Matrix Structure for Coupled Algorithms 119 Scalar-Implicit Coupling
• In scalar implicit coupling, components of m at P do not depend on each other. Thus, AP and AN is a diagonal tensor:
A =
• In this case, the “block system” represents 3 equations written together but not interacting: the block notation for the system is misleading for the level of coupling present in discretisation
• This leads towards a segregated method: we have three independent equa-tions written together. Lack of off-diagonal coefficients indicate the absence of component-to-component coupling
• Example of scalar coefficient terms: temporal derivative, diagonal and off-diagonal of convection and diffusion with scalar diffusivity
Block-Point Implicit Coupling
• In block-point implicit coupling the components of a vector variable m depend on each other in the same computational point, but each in-dividual component depends only of the neighbouring value of the same component
• Thus:
– In point P , mx depends on self, my and mz. Thus, the diagonal coefficient ap would be a full 3 × 3 matrix
AP =
– In the off-diagonal, mx fo location P will depend only on mx at N,
– As before, in most cases, the diagonal components are identical.
AN = a I (8.37)
The first form is typical for anisotropic porous media.
– In this situation, the “transport” part of the system (as depicted by AN exhibits segregated behaviour, combined by a point-coupled prob-lem for each computational point
Scalar-Point Vector-Implicit Coupling
• In the third combination, local point components of mx are decoupled, but the coupling to the neighbouring locations is complete. Thus
AP =
• Such cases are relatively rare and typically appear from tensorial diffusion problems and in some cases of rotational coupling
Full Block Coupling
• In full block coupling, each component of m depends on all other compo-nents both in the local and neighbouring computational points. Thus, both the diagonal and off-diagonal coefficient take full tensor form:
AP =
(note that component values will be different between the two)
8.4 Matrix Structure for Coupled Algorithms 121
• This is the most complex form of coupling, where “everything is related to everything else” [Lenin]
Composite Variables
• In some equations, the system will be coupled not only across the compo-nents of vectors and tensors, but also across different variables. In such cases, we may write a composite variable formulation, where all equations are grouped together into a single equation
• The fact that a composite variable is not a Cartesian tensor needs to be kept in mind. Calculation of gradients, divergence etc. is no longer trivial:
the physical meaning of the field needs to be taken into account
• Example: compressible Navier-Stokes equations
U =
• Note that U above holds 5 scalar values: 1 for the density, 3 momentum components (ρux, ρuy, ρuz) and one for energy
• This tactics makes sense only if the variables are strongly coupled to each other. Thus, full block coupling typically appears for such systems
Non-Linear Coupling
• Additional complications will arise for cases where the matrix coefficients are also a function of the solution: non-linearity
• Example: convection term in the momentum equation ∇•(u u). Here, com-ponents of AP and AN depend on the solution itself, thus creating a non-linear system
• Standard methods, line the Newton linearisation require the evaluation of the Jacobian, which is complex and costly. In reality, simple linearisation is used most often: evaluate AP and AN based on the current value of u and re-calculate u.
Saddle Block Systems
• A system of equations central to our interest (incompressible Navier-Stokes equations) has a worrying property: wrong equations!
– Unknowns: velocity vector u (3 vector components) and pressure p (scalar)
– Equations: momentum equation (3 vector components)
∂u
∂t + ∇•(uu) − ∇•(ν∇u) = −∇p (8.43)
– Continuity equation:
∇•u = 0 (8.44)
– Continuity equation sets a condition on velocity divergence ∇•u, which is a scalar – this makes is a scalar equation
– Formally, we have 1 vector equation and one vector unknown and one scalar equation
– . . . but the scalar equation is given in terms of u and not p!!!
• This kind of system is termed the saddle-point system: equations that govern p do not depend on it. Formally, we can write the system as follows:
· [Au] [∇(.)] Note the absence of entries for p in the diagonal matrix! Off diagonal blocks actually represent the discretised form of the gradient and divergence operator, multiplied by p and u, respectively. The diagonal block [Au] contains the discretised form of the momentum equation, excluding the pressure gradient term
• While there exists a large set of zero diagonal entries, this matrix can be solved. However, naive solution method would require a direct linear equa-tion solver, making it extremely expensive. We shall look for cheaper and faster solution methods
• In compressible flows, the density-pressure relationship replaces the zero diagonal block. However, as we approach the incompressibility limit, the system approaches the saddle point form