We present a novel a posteriori subcell finite volume limiter for high order discontinuous Galerkin (DG) finite element schemes for the solution of nonlinear hyperbolic PDE systems in multiple space dimensions on fixed and moving unstructured simplex meshes. The numerical method belongs to the family of high order fully discrete one-step ADER-DG schemes [12, 45] and makes use of an element-local space-time Galerkin finite element predictor. Our limiter is based on the MOOD paradigm, in which the discrete solution of the high order DG scheme is checked a posteriori against a set of physical and numerical admissibility criteria, in order to detect spurious oscillations or unphysical solutions and in order to identify the so-called troubled cells. Within the detected troubled cells the discrete solution is then discarded and recomputed locally with a more robust finite volume method on a fine subgrid.
In this work, we propose for the first time to use a high order ADER-CWENO finite volume scheme as subcell finite volume limiter on unstructured simplex meshes, instead of a classical second order TVD scheme. Our new numerical scheme has been developed both for fixed Eulerian meshes as well as for moving Lagrangian grids. It has been carefully validated against a set of typical benchmark problems for the compressible Euler equations of gas dynamics and for the equations of ideal magnetohydrodynamics (MHD).

Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.

In this paper we present a parallelization strategy on distributed memory systems for the Fast Kinetic Scheme — a semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] for solving kinetic equations. The original algorithm was proposed for the BGK approximation of the collision kernel. In this work we deal with its extension to the full Boltzmann equation in six dimensions, where the collision operator is resolved by means of fast spectral method. We present close to ideal scalability of the proposed algorithm on tera- and peta-scale systems.

The equioscillation principle is an important rule to fix the parameter for the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions. For parabolic PDEs with integer order temporal derivative, such a principle yields optimal Robin parameter, while in our previous study we found numerically that it is not always the case for time fractional PDEs: the Robin parameter determined by the equioscillation principle is sometimes far away from optimal. In this paper, by using the time fractional Cable equations as the model, we show that our previous finding does not happen occasionally but an inherent property of the SWR algorithm. Our analysis also reveals an essential difference between the asymptotic convergence rates in the overlapping and non-overlapping cases. Numerical results are provided to validate our theoretical analysis.

A new kinetic model is proposed to solve the Riemann problem for stiffened gas interface from piecewise linear discontinuous initial distributions. In the model gas particles on each side of the fluid interface are reflected back from the interface, respectively, which is moving with a velocity to achieve the force balance between both sides. Compared with the existing Riemann solver, the present model can keep second order accuracy in both space and time. It is also capable of eliminating the numerical mixing at the fluid interface which is different from existing kinetic models. The new model is applied to the numerical flux calculation at a cell interface and with the help of homogeneous equilibrium mixture assumption within a cell, a new gas-kinetic scheme is developed for multimaterial flow with stiffened equation of state. The scheme is tested with several typical high-speed multifluid flows, including the water-gas shock tube flows and the shock-water cylinder interaction. The computed results are in good agreement with other numerical and experimental studies. Fluid interfaces as well as shock waves are sharply captured, free of numerical oscillation near the interface, even for density ratio up to about one thousand, which validate its high accuracy, strong robustness and good parallel performance.

We analyze the effective time step size of a nonlinear convex splitting scheme for the Cahn–Hilliard (CH) equation. The convex splitting scheme is unconditionally stable, which implies we can use arbitrary large time-steps and get stable numerical solutions. However, if we use a too large time-step, then we have not only discretization error but also time-step rescaling problem. In this paper, we show the time-step rescaling problem from the convex splitting scheme by comparing with a fully implicit scheme for the CH equation. We perform various test problems. The computation results confirm the time-step rescaling problem and suggest that we need to use small enough time-step sizes for the accurate computational results.

Anion defect concentrations in the uranium dioxide were calculated in a wide temperature range by the molecular dynamics (MD) method, and a good agreement with the neutron scattering experimental data is shown. Groups of isolated interstitial anions with the absence of anion vacancies in N nearest neighborhoods (N=1−4) were extracted, formation energies of each group in the different phase states were derived.
The formation energy of anion defects 4eV, derived by the MD method at low temperatures, in general coincides with the experimental estimations (3–4)eV and the lattice static results 4.1eV, using the same interaction potential. At high temperatures in the superionic state anion Frenkel defects (with N=3,4) have negative formation energy of about (-14)eV, concentrations of these defects are very small compared with the total defect concentration, including non-stoichiometric crystals. The validity of the point defects model in the superionic state and transition regions is also discussed.
The model of anion diffusion via the formation of short-living pairs "vacancy-interstitial anion" is proposed and confirmed by MD calculations. It is shown that in the superionic state the exchange mechanism dominates, and at low temperatures it coexists with the diffusion via the formation of long-living anion Frenkel pairs.

The construction of high order accurate generalized finite difference method for inviscid compressible flows still remains an open problem in the literature. In this paper, the high order accurate generalized finite difference schemes have been developed based on the high order reconstruction and the high order numerical flux evaluation on a local cloud of points. The WBAP limiter based on the secondary reconstruction is used to suppress oscillations near discontinuities. The implementation of high order accurate boundary conditions is of critical importance in the construction of high order schemes. A new method is proposed for the high order accurate boundary treatment. Several standard test cases are solved to validate the accuracy, efficiency and shock capturing capability of the proposed high order schemes.

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean-field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.

In this paper, we develop the Hamiltonian conservative and $L^2$ conservative local discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) type equations with the minimal stencil. For the time discretization, we adopt the semi-implicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stability of the fully discrete schemes is provided by Fourier analysis for the linearized KdV equation. Numerical examples are shown to illustrate the capability of these schemes. Compared with the dissipative LDG scheme, the numerical simulations also indicate that the conservative LDG scheme with high order time discretization can reduce the long time phase error validly.

We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adjusting a parameter q leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.

Energetic Variational Approach is used to derive a novel thermodynamically consistent three-phase model of a mixture of Newtonian and visco-elastic fluids. The model which automatically satisfies the energy dissipation law and is Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard equations. Modified General Navier Boundary Condition with fluid elasticity taken into account is also introduced for using the model to study moving contact line problems. Energy stable numerical scheme is developed to solve system of model equations efficiently. Convergence of the numerical scheme is verified by simulating a droplet sliding on an inclined plane under gravity. The model can be applied for studying various biological or biophysical problems. Predictive abilities of the model are demonstrated by simulating deformation of venous blood clots with different visco-elastic properties and experimentally observed internal structures under different biologically relevant shear blood flow conditions.