The linearly implicit Lintrap scheme discussed in this paper is based on the trapezoidal formula. It has been used extensively in many applications. Lintrap is first order for nonautonomous problems, and its local truncation error doesn’t match the trapezoidal rule on which it is based. The author develops a new, linearly implicit scheme, also based on the trapezoidal rule. This new scheme is second order, and has the same local truncation error as the trapezoidal rule. Four nonlinear numerical examples demonstrate the rate of convergence, and its advantages over the Lintrap and Crank-Nicolson schemes.

The author has included examples from ordinary differential equations (ODEs) and partial differential equations (PDEs) to demonstrate the advantages of the method over the currently used Lintrap scheme. The paper is easy to read, and I have already recommended it to my numerical ODE students. I will certainly recommend its use in general. It is an improvement on the state-of-the-art solution for nonlinear differential equations.