### 11.1.2. The Coupled System of Equations

The linear set of equations that arise by applying the finite volume method to all elements in the domain are discrete conservation equations. The system of equations can be written in the form:

| (11–45)

where is the solution, b the right hand side, a the coefficients of the equation, i is the identifying number of the control volume or node in question, and nb means “neighbor”, but also includes the central coefficient multiplying the solution at the ith location. The node may have any number of such neighbors, so that the method is equally applicable to both structured and unstructured meshes. The set of these, for all control volumes constitutes the whole linear equation system. For a scalar equation (for example, enthalpy or turbulent kinetic energy), , and b i are each single numbers. For the coupled, 3D mass-momentum equation set, they are a (4 x 4) matrix or a (4 x 1) vector, which can be expressed as:

| (11–46)

and

| (11–47)

| (11–48)

It is at the equation level that the coupling in question is retained and at no point are any of the rows of the matrix treated any differently (for example, different solution algorithms for momentum versus mass). The advantages of such a coupled treatment over a non-coupled or segregated approach are several: robustness, efficiency, generality, and simplicity. These advantages all combine to make the coupled solver an extremely powerful feature of any CFD code. The principal drawback is the high storage needed for all the coefficients.

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