In this paper, new Locally Linearized Runge-Kutta formulas of Dormand and Prince are introduced with the purpose of integrating large systems of initial value problems. The new formulas are obtained by replacing the Padé approximation for matrix exponential by a Krylov-Padé approximation in the embedded formulas proposed in a previous work. Unlike other high order exponential integrators, these new formulas involve the approximation of just a single phi-function times vector, which makes them particularly attractive from a computational viewpoint. With these new formulas, adaptive schemes are constructed with novelties in the approximation of phi-functions times vectors by Krylov-Padé approximations, and in the strategies for controlling the approximation errors, for estimating the dimension of the Krylov subspaces, and for the reuse of the Jacobians. In addition, numerical experiments are performed to show the potential of the new embedded formulas and adaptive codes in the integration of known physical, biophysical, and physical-chemistry models.

在本文中，引入了新的局部线性化的Dormand和Prince的Runge-Kutta公式，目的是集成大型的初值问题系统。通过在先前工作中提出的嵌入公式中用Krylov-Padé逼近代替矩阵指数的Padé逼近来获得新公式。与其他高阶指数积分器不同，这些新公式仅涉及单个phi函数时间向量的逼近，这从计算角度来看使它们特别有吸引力。借助这些新公式，在利用Krylov-Padé近似逼近phi函数乘以向量以及在控制近似误差的策略，估算Krylov子空间的维数方面，构造了具有新颖性的自适应方案，为了雅各布主义者的重用 此外，进行了数值实验，以显示新的嵌入式公式和自适应代码在已知物理，生物物理和物理化学模型的集成中的潜力。