Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi's method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p = s in the multi-implicit case and order p = s − 1 in the singly-implicit case with arbitrary step sizes and s ≤ 5. Numerical tests in MATLAB for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.
We consider explicit two-step peer methods for the solution of nonstiff differential systems. By an additional condition a subclass of optimally zero-stable methods is identified that is superconvergent of order p = s + 1, where s is the number of stages. The new condition allows us to reduce the number of coefficients in a numerical search for good methods. We present methods with 4-7 stages which are tested in FORTRAN90 and compared with DOPRI5 and DOP853. The results confirm the high potential of the new class of methods.
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