Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge–Kutta (EFTDDIRK) methods for the numerical solution differential equations with oscillatory solutions. First, general format so-called modified (TDDIRK) is proposed. Their order conditions up to six are derived by introducing set bi-colored rooted trees deriving elementary weights. Next, build exponential fitting in these TDDIRK treat solutions, leading EFTDDIRK methods. particular, family 2-stage fourth-order, fifth-order, 3-stage sixth-order schemes derived. These can be considered as superconvergent The stability phase-lag analysis also investigated, optimized fourth-order schemes, which turn out much more accurate efficient than their non-optimized versions. Finally, carry experiments on some test problems. Our results clearly demonstrate accuracy efficiency newly when compared existing trigonometically/exponentially same literature.