A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods ($ABC$-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in $ABC$-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. $ABC$-schemes are $A$- and $L$-stable methods of the second order, but there are $ABC$-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of $ABC$-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.