3.24 Example of building a matrix equation
The previous sections describe methods to discretise derivative and other terms in order to build a matrix equation for a given physical equation. Let us demonstrate the construction of a matrix equation, using the momentum conservation equation from Sec. 3.23 as an example. It is a vector equation, so produces 3 matrix equations for , and .
The ﬁrst term, the time derivative , might be discretised with the Euler scheme Eq. (3.21 ). Matrix equations are constructed in extensive form as discussed in Sec. 3.6 . Hence, the contributions from Eq. (3.21 ) to matrix coeﬃcients and source vector are scaled by cell volume , i.e. and , respectively, as illustrated below.
The transported might be discretised using the linear upwind scheme described in Sec. 3.14 . The scheme ﬁrst applies upwind discretisation, which contributes outgoing positive ﬂuxes to diagonal coeﬃcients and negative ﬂuxes to oﬀ-diagonals. It then adds an explicit contribution based on an extrapolated gradient (see Sec. 3.14 ). The gradient is usually calculated by Eq. (3.18 ) with gradient limiting from Sec. 3.16 .