The generalized Euler process for exponentially dominant systems

In order to obtain numerical solutions for the exponentially dominant systems, Eulerʹs method is improved. The improved process is based on the use of the matrix exponential, but requires neither a repeated evaluation of a matrix exponential nor a replacement of the matrix exponential by a suitable approximant. It is shown that the process is explicit, convergent, of first order, and contractive for dissipative problems. It is also shown that the process can be efficiently implemented for the exponentially dominant systems having a long-time oscillatory behavior even with relatively large step sizes and is effective even for nondissipative systems. Numerical results are compared with other methods.