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An attempt to construct fast Runge–Kutta methods for delay differential equations with distributed delays is made. As a starting point functional continuous methods for general retarded functional differential equations are considered. New explicit methods of order three and four are constructed, which are more effective than functional continuous Runge–Kutta methods of the same order. Numerical tests confirm the convergence order of the new methods. Comparison to closely related Runge–Kutta methods for Volterra integro-differential equations is made.

A theorem was originally proposed to deal with the stochastic theta methods when they are applied to a linear test equation with delay and non-delay in a diffusion term. We extend the theorem to a proposition in a general form, and use it for stability analysis of stochastic orthogonal Runge-Kutta-Chebyshev methods when the methods are applied to the test equation., International Conference on Numerical Analysis and Applied Mathematics 2019 (ICNAAM-2019), 23–28 September, 2019, Rhodes, Greece

Two families of structural four-stage Runge–Kutta methods of order five are considered. The choice of the scheme with the best numerical properties in each family is justified. The comparison of classical and Runge–Kutta–Nystrom fifth order methods is shown and the better performance for some test problems is demonstrated.Two families of structural four-stage Runge–Kutta methods of order five are considered. The choice of the scheme with the best numerical properties in each family is justified. The comparison of classical and Runge–Kutta–Nystrom fifth order methods is shown and the better performance for some test problems is demonstrated.

Diagonally implicit Runge–Kutta type methods are considered for retarded functional differential equations. Methods are constructed to be L-stable when applied to ordinary differential equations. Two approached are used: complete diagonally implicit methods and methods with first explicit stage. It is shown that the latter lets construction of methods of desired orders with fewer stages.

We construct here an embedded Dormand–Prince pair of explicit methods of orders 6 and 4 for systems of ordinary differential equations with special structure, namely with two parts, in which the right-hand sides are dependent only on the unknown functions from the other group. The number of stages is six, which is fewer than for general explicit Runge–Kutta methods. The comparison to Dormand–Prince method of the same computation cost is made showing the higher accuracy of the suggested method.

We consider here numerical methods for systems of retarded functional differential equations of two equations in which the right-hand sides are cross-dependent of the unknown functions, i.e. the derivatives of unknowns don’t depend on the same unknowns. It is shown that using the special structure of the system one can construct functional continuous methods of Runge–Kutta type with fewer stages than it is necessary in case of general Runge–Kutta functional continuous methods. Order conditions and example methods of orders three and four are presented. Test problems are solved, demonstrating the declared convergence order of the new methods.

When solving delay differential equations (DDEs) with state-dependent delays the problem of breaking point detection is important. Points where the solution is not smooth enough to provide the order of the method must be included into the computational mesh, otherwise a reduction in the order of the solution will result. The problem, however is to detect and compute such points efficiently. Breaking points arise every time a delay falls on a previous breaking point (either of the calculated solution or in the history function). In the case of retarded DDEs the new breaking point is (at least) one order smoother than the previous breaking point that gave rise to it. For fixed or time-dependent delays the breaking points can be precomputed independent of the solution, but for state-dependent delays the positions of the breaking points depend on the computed solution. If a breaking point is detected and the step-size is changed in order to incorporate the point into the mesh, then the new step-size generates a new solution and the breaking point moves. Consequently, breaking point detection is traditionally performed iteratively, and is computationally expensive. The same breaking point can also be detected multiple times. In the current work we propose a fast non-iterative method for finding breaking points with sufficient precision to preserve the order of up to third or fourth order methods. Our method makes use of analytic continuation of the solution across breaking points (including possible breaking points in the initial history function), and we explain how we handle this carefully to attain the desired order. Test results are presented for Explicit Functional Continuous Runge–Kutta methods, showing that they retain their order of convergence when the solutions have breaking points.