The paper deals with stiff systems of differential equations. To solve this sort of system numerically is a diffult task. In spite of the fact that we come across stiff systems quite often in the common practice, a very interesting and promissing numerical method of solving systems of ordinary differential equations (ODE) based on Taylor series has appeared. The question was how to harness the ...

The paper deals with stiff systems of differential equations. To solve this sort of system numerically is a diffult task. In spite of the fact that we come across stiff systems quite often in the common practice, a very interesting and promissing numerical method of solving systems of ordinary differential equations (ODE) based on Taylor series has appeared. The question was how to harness the ...

This paper introduces two new numerical methods for integration of stiff ordinary differential equations. Following the idea of quantization based integration, i.e., replacing the time discretization by state quantization, the new methods perform first and second order backward approximations allowing to simulate stiff systems. It is shown that the new algorithms satisfy the same theoretical pr...

15 Acknowledgements 17 Introduction 19 1 Dispersive Partial Differential Equations and Integrable Systems 29 1.1 Linear Dispersive Partial Differential Equations . . . . . . . . . . . . 29 1.2 Semilinear Dispersive PDEs . . . . . . . . . . . . . . . . . . . . . . . 32 1.2.1 Equations of NLS Type. . . . . . . . . . . . . . . . . . . . . . 33 1.2.2 Equations of KdV Type. . . . . . . . . . . . . ....

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by sw...