solving ordinary differential equations we start with integrating the general ode utover time stepfrom tn to tnc ztnc / /dt tnc tn tn nc the goal of the computation is tnc ( )while tn (un is the most recently known value of the challenge with the integral is that the integrand involves the unknown between tn and tnc the integral can be approximated by the famous simpson' rule ztnc / /dt nc nc tn the problem with this formula is that we do not know nc unc tnc and nc unc tnc as only un is available and only can then readily be computed to proceedthe idea is to use various approximations for nc and nc based on using well-known schemes for the ode in the intervals oetn tnc and oetn tnc let us split the integral into four termsztnc fonc fqnc fnnc / /dt tn where fonc fqnc and fnnc are approximations to nc and nc that can uti lize already computed quantities for fonc we can simply apply an approximation to unc based on forward euler step of size fonc un tnc ( this formula provides prediction of nc so we can for fqnc try backward euler method to approximate unc fqnc un fonc tnc ( with fqnc as an approximation to nc we can for the final term fnnc use midpoint method (or central differencealso called crank-nicolson methodto approximate unc fnnc un fonc tnc ( we have now used the forward and backward euler methods as well as the centered difference approximation in the context of simpson' rule the hope is that