TASE operators to stabilize explicit parallelizable peer methods

In this talk we derive a new class of linearly implicit numerical methods for stiff initial value problems, also mentioning possible strategies for building software for the related parallel implementation. The proposed numerical schemes are obtained by combining parallelizable explicit peer methods and Time-Accurate and Highly-Stable Explicit (TASE) operators. TASE operators, which are matrices, involving the exact Jacobian, to be premultiplied by the vector field of the problem considered, have recently been introduced to stabilize explicit Runge-Kutta (RK) methods with s stages and order p=s, s=2, 3, 4. These matrices depend on some free parameters to be set suitably to achieve the desired accuracy and stability properties for the TASE-RK methods thus obtained. We show that for appropriate choices of the free parameters, the combination of TASE operators and parallelizable explicit peer methods leads to a new class of linearly implicit parallelizable peer methods with good accuracy and stability [3]. Furthermore, inspired by exponential integrators, we 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 show that the new TASE-peer schemes are competitive with TASE-RK methods and other well-known stiff integrators, being potentially more advantageous due to the related parallel structure. Finally, notes on the construction of software for the parallelization of TASE-peer methods in CUDA environment are provided, with hints at future developments by using the Matlab PCT (Parallel Computing Toolbox).