Time-accurate and highly-stable explicit peer methods for stiff differential problems

We derive a new class of parallelizable two-step peer methods for the numerical solution of stiff systems of Ordinary Differential Equations (ODEs), inspired by a technique introduced in Bassenne et al., (2021) and Calvo et al., (2021). It consists in the improvement of the stability properties of explicit Runge–Kutta (RK) methods by premultiplying the vector field of the problem to solve by a carefully chosen family of operators, called Time-Accurate and Highly-Stable Explicit (TASE) operators. In this work, we show that for suitable choices of the free parameters of the family of TASE operators proposed in Calvo et al., (2021), it is possible to improve the stability properties of explicit parallelizable peer methods. Furthermore, we also introduce a new family of TASE operators that fits better than the existing ones on some problems of interest in applications. Numerical tests on Partial Differential Equations (PDEs) are carried out to show that the new TASE-peer schemes are competitive with TASE-RK methods and other well-known stiff integrators, and are potentially more advantageous.