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A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme.

The authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case.

Numerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.

This paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

This paper seeks to develop an adaptive finite volume algorithm, and to present an extensive numerical analysis of it.

The effectiveness of the developed algorithm is demonstrated through practical and computationally challenging problems. The algorithm is tested for a wide range of singularities.

The convergence of the presented algorithm is independent of the regularity of the problems. It is shown that the our algorithm produces more accurate and well conditioned matrix systems.

Though the presented algorithm works for extreme singularities on rectangular meshes, it may not be as efficient if the underlying meshes are distorted, and it may not converge. Further research is under way for including the multi‐point approximation technique into the algorithm.

Almost all reservoir simulators use the two‐point method, and this algorithm is based on this method. The algorithm can be easily incorporated into the reservoir simulators. The results show that such an implementation will greatly improve the computational efficiency of the simulators. The work is useful for computational scientists, and especially for the researchers in oil industries. The paper reports the numerical work with practical applications.

The paper develops an adaptive finite volume algorithm. It is shown that adaptive meshes represent the underlying problem more accurately, and matrix systems associated with adaptive meshes are easier to solve compared with matrix systems associated with uniform meshes.

The purpose of this paper is to develop and apply a high-order numerical method based on finite volume approximation for quadratic integrate-and-fire (QIF) neuron model with the help of population density approach.

The authors apply the population density approach for the QIF neuron model to derive the governing equation. The resulting mathematical model cannot be solved with existing analytical or numerical techniques owing to the presence of delay and advance. The numerical scheme is based along the lines of approximation: spatial discretization is performed by weighted essentially non-oscillatory (WENO) finite volume approximation (FVM) and temporal discretization are performed by strong stability-preserving explicit Runge–Kutta (SSPERK) method. Compared with existing schemes of orders 2 and 3 from the literature, the proposed scheme is found to be more efficient and it produces accurate solutions with few grid cells. In addition to this, discontinuity is added in the application of the model equation to illustrate the high performance of the proposed scheme.

The developed scheme works nicely for the simulation of the resulting model equation. The authors discussed the role of inhibitory and excitatory parts in variation of neuronal firing. The validation of the designed scheme is measured by its comparison with existing schemes in the literature. The efficiency of the designed scheme is demonstrated via numerical simulations.

It is expected that the present study will be a useful tool to tackle the complex neuron model and related studies.

The novel aspect of this paper is the application of the numerical methods to study the modified version of leaky integrate-and-fire neuron based on a QIF neuron. The model of the current study is inspired from the base model given in Stein (1965) and modified version in Kadalbajoo and Sharma (2005) and Wang and Zhang (2014). The applicability was confirmed by taking some numerical examples.

The paper presents a numerical technique for the simulation of the effects of grey‐diffusive surface radiation on fluid flow using a finite volume procedure for two‐dimensional (plane and axi‐symmetric) geometries. The governing equations are solved sequentially, and the non‐linearities and coupling of variables are accounted for through outer iterations (coefficients updates). In order to reduce the number of outer iterations, a multigrid algorithm was implemented. The radiating surface model assumes a non‐participating medium, semi‐transparent walls and constant elementary surface temperature and radiation fluxes. The calculation of view factors is based on the analytical evaluation for the plane geometry and numerical integration for axi‐symmetric geometry. Ashadowing algorithm was implemented for the calculation of view factors in general geometries. The method for the calculation of view factors was first tested by comparison with available analytical solutions for a complex geometric configuration. The flow prediction code combined with radiation heat transfer was verified by comparisons with analytical one‐dimensional solutions. Further test calculations were done for the flow and heat transfer in a cavity with a radiating submerged body. As an example of the capabilities of the method, transport processes in metalorganic chemical vapour deposition (MOCVD) reactors were simulated.

The extrudate swell phenomenon is analysed by solving, simultaneously, the Navier‐Stokes equations along with the continuity equation by means of a finite volume method. In this work, the planar jet flows of incompressible viscous Newtonian and power‐law fluids for Reynolds numbers as high as 75 are simulated. The method uses the velocity components and pressure as the primitive variables and employs an unstructured triangular grid and triangular or polygonal control volume for each separate variable. The numerical results show good agreement with previously reported experimental and numerical results. Shear thickening results in an increase in swelling ratio, while the introduction of surface tension results in a describes in swelling ratio.

A mixed implicit semi Lagrangian finite difference‐finite volume method for numerical simulation of 2D air motion inside cylinders is derived and discussed. A conformal mapping from a physical (moving) domain to a computational (fixed) one is resorted in order to deal with a grid independent of time, making the numerical code very efficient. The numerical method is mass and energy conservative, unconditionally stable and at each timestep requires the solution of two well structured five‐band systems of linear equations. Its accuracy is first order in time and second one in space where the solution is smooth, while due to FCT space accuracy drops to the first order where the solution is steep. Stability of the method is proved both by a classical Von Newmann analysis and analysis of the matrices involved in the systems of linear equations. All these elements make the numerical method particularly fast. Numerical experiments are performed that show the influence of the maximum Courant number (with respect to the fluid speed) on the performance of the numerical method; moreover, comparison of simulations with a major existing code for engines is worked out.

The use of the prominent finite difference time‐domain (FDTD) method for the time‐domain solution of electromagnetic wave propagation past devices with small geometrical details can require very fine grids and can lead to unmanageable computational time and storage. The purpose of this paper is to extend the analysis of a discontinuous Galerkin time‐domain (DGTD) method (able to handle possibly non‐conforming locally refined grids, based on portions of Cartesian grids) and investigate the use of perfectly matched layer regions and the coupling with a fictitious domain approach. The use of a DGTD method with a locally refined, non‐conforming mesh can help focusing on these small details. In this paper, the adaptation to the DGTD method of the fictitious domain approach initially developed for the FDTD is considered, in order to avoid the use of a volume mesh fitting the geometry near the details.

Based on a DGTD method, a fictitious domain approach is developed to deal with complex and small geometrical details.

The fictitious domain approach is a very interesting complement to the FDTD method, since it makes it possible to handle complex geometries. However, the fictitious domain approach requires small volume elements, thus making the use of the FDTD on wide, regular, fine grids often unmanageable. The DGTD method has the ability to handle easily locally refined grids and the paper shows it can be coupled to a fictitious domain approach.

Although the stability and dispersion analysis of the DGTD method is complete, the theoretical analysis of the fictitious domain approach in the DGTD context is not. It is a subject of further investigation (which could provide important insights for potential improvements).

This is believed to be the first time a DGTD method is coupled with a fictitious domain approach.

We consider, as discretization in space, the nonconforming mesh developed in SUSHI (Scheme Using Stabilization and Hybrid Interfaces) developed in Eymard et al. (2010) for a semi-linear heat equation. The time discretization is performed using a uniform mesh. We are concerned with a nonlinear scheme that has been studied in Bradji (2016) in the context of the general framework GDM (Gradient Discretization Method) (Droniou et al., 2018) which includes SUSHI. We provide sufficient conditions on the size of the spatial mesh and the time step which allow to prove a W1,∞(L2)-error estimate. This error estimate can be viewed as an improvement for the W1,2(L2)-error estimate proved in Bradji (2016). The W1,∞(L2)-error estimate we want to prove in this note was stated without proof in Bradji (2016, Remark 7.2, Page 1302). Its proof is based on a comparison with an appropriately chosen auxiliary finite volume scheme along with the derivation of some new estimates on its solution.

A finite volume, semi‐implicit scheme, is proposed and discussed, which solves the two‐dimensional Euler equations for the hypersonic flow of a mixture of chemically reactive specie. The present scheme can be applied on a general, unstructured grid. The first order version guarantees non negativity of the densities and of the vibrational energies for arbitrarily large time steps. The semi‐implicit time discretization of advective terms and the fully implicit discretization of the highly nonlinear terms yield a simple and efficient computer algorithm. Numerical tests show that shocks are well captured and the correct profiles for the chemical specie are reproduced at low computational cost.

This paper aims to extend the hybrid atomistic-continuum multiscale method developed by Vu et al. (2016) to study the gas flow problems in long microchannels involving density variations.

The simulation domain is decomposed into three regions: the bulk where the continuous Navier–Stokes and energy equations are solved, the neighbourhood of the wall simulated by molecular dynamics and the overlap region which connects the macroscopic variables (density, velocity and temperature) between the two former regions. For the simulation of long micro/nanochannels, a strategy with multiple molecular blocks all along the fluid/solid interface is adopted to capture accurately the macroscopic velocity and temperature variations.

The validity of the hybrid method is shown by comparisons with a simplified analytical model in the molecular region. Applications to compressible and condensation problems are also presented, and the results are discussed.

The hybrid method proposed in this paper allows cost-effective computer simulations of large-scale problems with an accurate modelling of the transfers at small scales (velocity slip, temperature jump, thin condensation films, etc.).