We investigate two-photon double ionization of helium by intense $(\approx {10}^{15}\mathrm{W}∕{\mathrm{cm}}^2)$ ultrashort $(\approx 300\mathrm{as})$ soft-x-ray pulses $(\mathrm{E}=91.6\mathrm{eV})$. The time-dependent two-electron Schrödinger equation is solved using a coupled channel method. We show that for ultrashort pulses the angular distribution of ejected electrons depends on the pulse duration and provides insight into the role of electron correlations in the two-electron photoemission process. The angular distribution at energies near the “independent-electron” peaks is close to dipolar while it acquires in the “valley” of correlated emission a significant quadrupolar component within a few hundred attoseconds.

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 10⁵ different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics.

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker–Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge–Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Steam condensation induced water hammer (CIWH) phenomena are investigated and new theoretical results are presented. We use the WAHA3 model based on two-phase flow six first-order partial differential equations that present one dimensional, surface averaged mass, momentum and energy balances. A second order accurate high-resolution shock-capturing numerical scheme was applied with different kind of limiters in the numerical calculations. The applied two-fluid model shows some similarities to RELAP5 which is widely used in the nuclear industry to simulate nuclear power plant accidents. This model was validated with different CIWH experiments which were performed in the PMK-2 facility, which is a full-pressure thermohydraulic model of the nuclear power plant of VVER-440/312 type in the Energy Research Center of the Hungarian Academy of Sciences and in the Rosa facility of the Japan Atomic Energy Agency. In our present study we show the first part of a planned large database which will give us the upper and lower flooding mass flow rates for various pipe geometries where CIWH can happen. Such a reliable database would be a great help for future reactor constructions and scheming.

We investigate the incompressible and compressible heat conducting boundary layer with applying the two-dimensional self-similar Ansatz. Analytic solutions can be found for the incompressible case which can be expressed with special functions. The parameter dependencies are studied and discussed in details. In the last part of our study we present the ordinary differential equation (ODE) system which is obtained for compressible boundary layers.