25 - Order Conditions for Linear Multistep Methods

Order Conditions for Linear Multistep Methods

Linear Multi-Step method has consistency order $p$ if and only if

$$\sigma (1)\ne 0\text{ and }\sum _{j=0}^k{\alpha }_j=0\text{ and }\sum _{j=0}^k{\alpha }_j{j}^q=q\sum _{j=0}^k{\beta }_j{j}^{q-1}\text{ for }q=1,\cdots ,p$$

where it is consistent if it is satisfied for at least $p=1$

Consistent

Consistent if

$$\rho (1)=0\text{ and }{\rho }^{\prime }(1)=\sigma (1)\ne 0$$