A theoretical solution of the Riemann problem to the two-phase flow model in non-conservative form of Saurel and Abgrall is presented under the assumption that all the nonlinear waves are shocks. The solution, called 4-shock Riemann solver, is then utilized to construct a path-conservative scheme for numerical solution of a general initial boundary value problem for the two-phase flow model in the non-conservative form.
Moreover, a high-order path-conservative scheme of Godunov type is given via the MUSCL reconstruction and the Runge-Kutta technique first in one dimension, based on the 4-shock Riemann solver, and then extended to the two-dimensional case by dimensional splitting. A number of numerical tests are carried out and numerical results demonstrate the accuracy and robustness of our scheme in the numerical solution of the five-equations model for two-phase flow.

A fully three-dimensional numerical study of the dynamics and field-induced deformation of a sheared, superparamagnetic ferrofluid droplet immersed in a Newtonian viscous fluid is presented. The system is a three-dimensional, periodic channel with top and bottom walls displaced to produce a constant shear rate and with an external, uniform magnetic field perpendicular to the walls. The model consists of the incompressible Navier-Stokes equations with the extra magnetic stress coupled to the static Maxwell's equations. The coupled system is solved with unprecedented resolution and accuracy using a fully adaptive, Immersed Boundary Method. For small droplet distortions, the numerical results are compared and validated with an asymptotic theory. For moderate and strong applied fields, relative to surface tension, and weak flows a large field-induced droplet deformation is observed. Moreover, it is found that the droplet distortion in the vorticity direction can be of the same order as that occurring in the shear plane. This study highlights the importance of the three-dimensional character of a problem of significant relevance to applications, where a dispersed magnetic phase is employed to control the rheology of the system.

Many production and commercial unstructured CFD codes provide no better than 2nd-order spatial accuracy. Unlike structured grid procedures where there is an implied structured connectivity between neighboring grid points, for unstructured grids, it is more difficult to compute higher derivatives due to a lack of explicit connectivity beyond the first neighboring cells. Our goal is to develop a modular high-order scheme with low dissipation flux difference splitting that can be integrated into existing CFD codes for use in improving the solution accuracy and to enable better prediction of complex physics and noise mechanisms and propagation. In a previous study, a 3rd-order U-MUSCL scheme using a successive differentiation method was derived and implemented in FUN3D. Verification studies of the acoustic benchmark problems showed that the new scheme can achieve up to 4th-order accuracy. Application of the high-order scheme to acoustic transport and transition-to-turbulence problems demonstrated that with just 10% overhead, the solution accuracy can be dramatically improved by as much as a factor of eight. This paper examines the accuracy of the high-order scheme for turbulent flow over single and tandem cylinders. Considerably better agreement with experimental data is observed when using the new 3rd-order U-MUSCL scheme.

Nowadays equilibrium thermodynamic properties of materials can be obtained very efficiently by numerical simulations. If the properties are needed over a range of temperatures it is highly efficient to determine the density of states first. For this purpose histogram- and matrix-based methods have been developed. Here we present a performance comparison of a number of those algorithms. The comparison is based on three different benchmarks, which cover systems with discrete and continuous state spaces. For the benchmarks the exact density of states is known, for one benchmark – the FAB system – the exact infinite temperature transition matrix Q is also known. In particular the Wang-Landau algorithm in its standard and 1/t variant are compared to Q-methods, where estimates of the infinite temperature transition matrix are obtained by random walks with different acceptance criteria. Overall the Q-matrix methods perform better or at least as good as the histogram methods. In addition, different methods to obtain the density of states from the Q-matrix and their efficiencies are presented.

Here, a physical dispersion relation preserving (DRP) scheme has been developed by combined optimization of the spatial and the multi-stage temporal discretization scheme to solve acoustics problems accurately. The coupled compact difference scheme (CCS) has been spectrally optimized (OCCS) for accurate evaluation of the spatial derivative terms. Next, the combination of the OCCS scheme and the five stage Runge-Kutta time integration (ORK5) scheme has been optimized to reduce numerical diffusion and dispersion error significantly. Proposed OCCS−ORK5 scheme provides accurate solutions at considerably higher CFL number. In addition, ORK5 time integration scheme consists of low storage formulation and requires less memory as compared to the traditional Runge-Kutta schemes. Solutions of the model problems involving propagation, reflection and diffraction of acoustic waves have been obtained to demonstrate the accuracy of the developed scheme and its applicability to solve complex problems.

We are concerned with the profile reconstruction of a penetrable grating from scattered waves measured above the periodic structure. The inverse problem is reformulated as an optimization problem, which consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed problem. A Tikhonov regularization method and a Landweber iteration strategy are applied to the objective function to deal with the ill-posedness and nonlinearity. We propose a self-consistent method to recover a potential function and an approximation of grating function in each iterative step. Some details for numerical implementation are carefully discussed to reduce the computational efforts. Numerical examples for exact and noisy data are included to illustrate the effectiveness and the competitive behavior of the proposed method.

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172–1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.

In fluid film lubrication investigations, the homogenized Reynolds equation is used as an averaging model to deal with microstructures induced by rough or textured surfaces. The objective is a reduction of computation time compared to directly solving the original Reynolds equation which would require very fine computational grids. By solving cell problems on the microscale, homogenized coefficients are computed to set up a homogenized problem on the macroscale. For the latter, the discretization can be chosen much coarser than for the original Reynolds equation. However, the microscale cell problems depend on the macroscale film thickness and thus become parameter-dependent. This requires a large number of cell problems to be solved, contradicting the objective of accelerating simulations. A reduced basis method is proposed which significantly speeds up the solution of the cell problems and the computation of the homogenized coefficients without loss of accuracy. The suitability of both the homogenization technique and the combined homogenization/reduced basis method is documented for the application to textured journal bearings. For this purpose, numerical results are presented where deviations from direct solutions of the original Reynolds equation are investigated and the reduction of computational cost is measured.

A Ciarlet-Raviart type mixed finite element approximation is constructed and analyzed for a class of fourth-order elliptic problems arising from solving various gradient systems. Optimal error estimates are obtained, using a super-closeness relation between the finite element solution and the Ritz projection of the PDE solution. Numerical results agree with the theoretical analysis.

Using the idea of weighted and shifted differences, we propose a novel finite difference formula with second-order accuracy for the tempered fractional derivatives. For tempered fractional diffusion equations, the proposed finite difference formula yields an unconditionally stable scheme when an implicit Euler method is used. For the numerical simulation and as an application, we take the CGMYe model as an example. The numerical experiments show that second-order accuracy is achieved for both European and American options.

We present a micro-scale model describing the dynamics of pore water phase transition and associated mechanical effects within water-saturated soil subjected to freezing conditions. Since mechanical manifestations in areas subjected to either seasonal soil freezing and thawing or climate change induced thawing of permanently frozen land may have severe impacts on infrastructures present, further research on this topic is timely and warranted.
For better understanding the process of soil freezing and thawing at the field-scale, consequent upscaling may help improve our understanding of the phenomenon at the macro-scale.
In an effort to investigate the effect of the pore water density change during the propagation of the phase transition front within cooled soil material, we have designed a 2D continuum micro-scale model which describes the solid phase in terms of a heat and momentum balance and the fluid phase in terms of a modified heat equation that accounts for the phase transition of the pore water and a momentum conservation equation for Newtonian fluid. This model provides the information on force acting on a single soil grain induced by the gradual phase transition of the surrounding medium within a nontrivial (i.e. curved) pore geometry. Solutions obtained by this model show expected thermal evolution but indicate a non-trivial structural behavior.

This paper develops an enhanced finite element method for approximating a class of variational problems which exhibits the $Lavrentiev$ $gap$ $phenomenon$ in the sense that the minimum values of the energy functional have a nontrivial gap when the functional is minimized on the spaces $W^{1,1}$ and $W^{1,∞}$. To remedy the standard finite element method, which fails to converge for such variational problems, a simple and effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs the gap phenomenon by capping the derivatives of its input on a scale of $\mathcal{O}$($h^{−α}$) (where $h$ denotes the mesh size) for some positive constant $α$. A sufficient condition is proposed for determining the problem-dependent parameter $α$. Extensive 1-D and 2-D numerical experiment results are provided to show the convergence behavior and the performance of the proposed enhanced finite element method.