## 1.5.1 Consistency

As given in Section 1.3.4, an \(s\)-step multi-step method can be written as,

where the forcing terms have been moved to the left-hand side. Substituting the exact solution, \(u(t)\), into the left-hand side will produce a remainder which is in fact the opposite of the truncation error (see Equation 1.49),

If we only require that \(\tau \rightarrow 0\) (i.e. \(\tau = O({\Delta t})\)) as \({\Delta t}\rightarrow 0\), the method will not generally be consistent with the ODE. To see why, note that in the limit of \({\Delta t}\rightarrow 0\), the forcing terms will vanish since they are scaled by \({\Delta t}\). Thus, \(\tau \rightarrow 0\) would place a constraint only on the \(\alpha\)'s. Let's look at that constraint on the \(\alpha\)'s to build some insight. Substituting Taylor series about \(t=t^ n\) for the values of \(u\) gives,

Thus, for \(\tau \rightarrow 0\) as \({\Delta t}\rightarrow 0\) requires,

This constraint can be interpreted as requiring a constant solution, i.e. \(u(t) =\) constant, to be a valid solution of the multi-step method. Clearly, this is not enough to guarantee consistency with the ODE since the ODE requires \(u_ t = f(u,t)\).

To achieve a consistent discretization, we force \(\tau /{\Delta t}\rightarrow 0\) (i.e. \(\tau = O({\Delta t}^2)\)). This stronger constraint can be shown to enforce that the ODE is satisfied in the limit of \({\Delta t}\rightarrow 0\):

Thus, in the limit as \({\Delta t}\rightarrow 0\), to obtain \(\tau /{\Delta t}\rightarrow 0\) then the slope of the numerical method (i.e. the first term) must be equal to the forcing at \(t^ n\). In other words, the multi-step discretization would satisfy the governing equation in the limit. An equivalent way to write this consistency constraint is,

In terms of the local accuracy, consistency requires that the multi-step method be at least first-order \((p=1)\) since \(\tau = O({\Delta t}^{p+1})\) and consistency requires that \(\tau /{\Delta t}= O({\Delta t}^ p)\) must go to zero (i.e. \(p \geq 1\)).

**Exercise** Which of the following numerical methods is consistent?