On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients

Abstract

In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

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Acknowledgements

The authors like to thank Wolf-Jürgen Beyn for very helpful comments on non-autonomous evolution equations and Rico Weiske for good advice on programming. Also we like to thank two anonymous referees for their valuable suggestions. This research was partially carried out in the framework of Matheon supported by Einstein Foundation Berlin. ME would like to thank the Berlin Mathematical School for the financial support. RK also gratefully acknowledges financial support by the German Research Foundation (DFG) through the research unit FOR 2402—Rough paths, stochastic partial differential equations and related topics—at TU Berlin.

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Eisenmann, M., Kovács, M., Kruse, R. et al. On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients. Found Comput Math 19, 1387–1430 (2019). https://doi.org/10.1007/s10208-018-09412-w

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Keywords

  • Monte Carlo method
  • Evolution equations
  • Ordinary differential equations
  • Backward Euler method
  • Galerkin finite element method

Mathematics Subject Classification

  • 65C05
  • 65L05
  • 65L20
  • 65M12
  • 65M60