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In this paper, we provide high order hybrid asymptotic augmented finite volume schemes on a uniform grid for nonlinear weakly degenerate and strongly degenerate wave equations. The whole domain is divided into singular and regular subdomains by introducing an intermediate point. Puiseux series asymptotic technique is used in singular subdomain, and augmented numerical method is used in regular subdomain. The key to the method are the recovery of Puiseux series for the nonlinear degenerate wave equation in singular subdomain and the organic combination between the singular and regular subdomains by means of augmented variables related to singularity. In particular, through imposing a condition at the intermediate point, we can not only improve the accuracy of the augmented variables, but also avoid the restriction conditions when the mesh is divided in regular subdomain. The advantage of this method is that the global convergence order of the degenerate wave equation is determined by the augmented numerical scheme in regular subdomain. A rigorous error estimate is conducted for the solution of the degenerate wave equation. Numerical examples on weakly degenerate and strongly degenerate problems are provided to illustrate the effectiveness of the proposed method. Especially, we use the method to solve an interesting example of a degenerate wave equation with coefficient blow-up.

In some previous works, the authors have introduced a strategy to develop well-balanced high-order numerical methods for nonconservative hyperbolic systems in the framework of path-conservative numerical methods. The key ingredient of these methods is a well-balanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. In this article, the specific case of 1d systems of balance laws is addressed and difficulties are gradually introduced: the methods are presented in the simpler case in which the source term does not involve Dirac masses. Next, systems whose source term involves the derivative of discontinuous functions are considered. In this case, the notion of weak solution is discussed and the Generalized Hydrostatic Reconstruction technique is used for the treatment of singular source terms. A technique to preserve the well-balancedness of the methods in the presence of numerical integration is introduced. The strategy is applied to derive first, second and third order well-balanced methods for Burgers’ equation with a nonlinear source term and for the Euler equations with gravity.

We present a dissipative measure-valued (DMV)-strong uniqueness result for the compressible Navier–Stokes system with potential temperature transport. We show that strong solutions are stable in the class of DMV solutions. More precisely, we prove that a DMV solution coincides with a strong solution emanating from the same initial data as long as the strong solution exists. As an application of the DMV-strong uniqueness principle we derive a priori error estimates for a mixed finite element-finite volume method. The numerical solutions are computed on polyhedral domains that approximate a sufficiently a smooth bounded domain, where the exact solution exists.

The numerical investigation of magneto-hydrodynamic (MHD) mixed convection flow and entropy formation of non-Newtonian Bingham fluid in a lid-driven wavy square cavity filled with nanofluid was investigated by the finite volume method (FVM). The numerical data-based temperature and nanoparticle size-dependent correlations for the Al2O3-water nanofluids are used here. The physical model is a two-dimensional wavy square cavity with thermally adiabatic horizontal boundaries, while the right and left vertical walls maintain a temperature of TC and TH, respectively. The top wall has a steady speed of u=u0. Pertinent non-dimensional parameters such as Reynolds number (Re=10,100,200,400), Hartmann number (Ha=0,10,20), Bingham number (Bn=0,2,5,10,50,100,200), nanoparticle volume fraction (ϕ=0,0.02,0.04), and Prandtl number (Pr=6.2) have been simulated numerically. The Richardson number Ri is calculated by combining the values of Re with a fixed value of Gr, which is the governing factor for the mixed convective flow. Using the Response Surface Methodology (RSM) method, the correlation equations are obtained using the input parameters for the average Nusselt number (Nu¯), total entropy generation (Es)t, and Bejan number (Beavg). The interactive effects of the pertinent parameters on the heat transfer rate are presented by plotting the response surfaces and the contours obtained from the RSM. The sensitivity of the output response to the input parameters is also tested. According to the findings, the mean Nusselt numbers (Nu¯) drop when Ha and Bn are increased and grow when Re and ϕ are augmented. It is found that (Es)t is reduced by raising Ha, but (Es)t rises with the augmentation of ϕ and Re. It is also found that the ϕ and Re numbers have a positive sensitivity to the Nu¯, while the sensitivity of the Ha and Bn numbers is negative.

Purpose The purpose of this study is to compare various approximations method such as finite element, finite volume and finite difference approximations for the coupled diffusion problem with interface conditions. The property of discrete energy conservation and stability analysis are the main contribution of this work for coupling problem. Two basic couplings (Dirichlet-Neumann and heat flux coupling) are considered along with the coupled diffusion equations. Design/methodology/approach The Crank–Nicolson scheme of finite volume, finite element and finite difference methods are used. Findings The authors show that conservation is preserved in the case of DN-coupling with one-sided approximation in both finite difference and finite volume schemes. However, heat flux is maintained the conservation by the central difference approximations along coupled diffusion equations. Originality/value This is the authors’ original work and modification of the previously published work mentioned in the paper.

In this paper, a penalty-based cell vertex finite volume method (P-CV-FVM) is proposed for the computation of two-dimensional contact problems. The deformation of objects during contact is described using the Total Lagrangian momentum equation. The governing equations are discretized using the cell vertex finite volume method. The control volume is constructed around each grid node to facilitate the efficient and accurate calculation of contact stress using penalty functions. By analyzing a classic contact example, the appropriate range of scaling factors in the penalty function method is obtained. Multiple contact problems are calculated and the results are compared with those from the finite element method (FEM). The results indicate that a stable and accurate solution can only be obtained with a scaling factor range of 10 3 –10 12 under this method. In addition, the mesh convergence of this method is better than that of FEM, and it meets the computational accuracy of Hertz contact and frictional contact problems.

Reactive transport modelling (RTM) is a powerful tool for understanding subsurface systems where fluid flow and chemical reactions occur simultaneously. RTM has been widely used to understand the formation of dolomite by replacement of calcite, which can be an important control on carbonate reservoir quality. Dolomitisation is a reactive transport process governed by slow dolomite precipitation and cannot be correctly simulated without a kinetic rate model. The new CSMP++GEM coupled RTM code uses the GEMS3K kernel for solving geochemical equilibria by the Gibbs energy minimisation method with the CSMP++ framework that implements a hybrid finite element–finite volume method to solve partial differential equations. The unique feature of the new coupling is the mineral reaction kinetics, implemented via additional metastability constraints. CSMP++GEM is able to simulate single-phase flow and solute transport in porous media together with chemical reactions at different pressure, temperature, and water salinity conditions. This RTM assures mass conservation which is crucial when simulating transport of solutes with low concentrations over geological time. A full feedback of mineral dissolution/precipitation on the fluid flow is provided via corresponding porosity/permeability evolution and two source terms in the pressure equation. First, the mass source term accounts for the mass of solutes released during mineral dissolution or taken from the solution by mineral precipitation. The second source term attributes to the fact that the solution density is affected by mineral dissolution/precipitation, too. This effect is included through the equivalent water salinity, which is calculated from the total amount of dissolved solutes and is used to update the properties of saline water from the H 2 O–NaCl equation of state. This paper puts emphasis on the thorough mathematical derivation of the governing equations and a detailed description of the numerical solution procedure. Two sets of benchmarking results are presented. The first benchmark is a well-known 1D model of dolomitisation by MgCl 2 solution with thermodynamic reactions. In the second benchmark, CSMP++GEM is compared with TOUGHREACT on a 1D model of dolomitisation by sea water taking into account mineral reaction kinetics. The results presented in this paper demonstrate the ability of the CSMP++GEM code to correctly reproduce dolomitisation effects.

The Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.

We present a new hybrid semi-implicit finite volume / finite element numerical scheme for the solution of incompressible and weakly compressible media. From the continuum mechanics model proposed by Godunov, Peshkov and Romenski (GPR), we derive the incompressible GPR formulation as well as a weakly compressible GPR system. As for the original GPR model, the new formulations are able to describe different media, from elastoplastic solids to viscous fluids, depending on the values set for the model’s relaxation parameters. Then, we propose a new numerical method for the solution of both models based on the splitting of the original systems into three subsystems: one containing the convective part and non-conservative products, a second subsystem for the source terms of the distortion tensor and thermal impulse equations and, finally, a pressure subsystem. In the first stage of the algorithm, the transport subsystem is solved by employing an explicit finite volume method, while the source terms are solved implicitly. Next, the pressure subsystem is implicitly discretised using continuous finite elements. This methodology employs unstructured grids, with the pressure defined in the primal grid and the rest of the variables computed in the dual grid. To evaluate the performance of the proposed scheme, a numerical convergence analysis is carried out, which confirms the second order of accuracy in space. A wide range of benchmarks is reproduced for the incompressible and weakly compressible cases, considering both solid and fluid media. These results demonstrate the good behaviour and robustness of the proposed scheme in a variety of scenarios and conditions.

In this paper, the safety and thermal comfort of protective clothing used by firefighters was analyzed. Three-dimensional geometry and morphology models of real multilayer assemblies used in thermal protective clothing were mapped by selected Computer-Aided Design (CAD) software. In the designed assembly models, different scales of the resolution were used for the particular layers – a homogenization for nonwoven fabrics model and designing the geometry of the individual yarns in the model of woven fabrics. Then, the finite volume method to simulate heat transfer through the assemblies caused by their exposure to the flame was applied. Finally, the simulation results with experimental measurements conducted according to the EN ISO 9151 were compared. Based on both the experimental and simulation results, parameters describing the tested clothing protective features directly affecting the firefighter’s safety were determined. As a result of the experiment and simulations, comparable values of these parameters were determined, which could show that used methods are an efficient tool in studying the thermal properties of multilayer protective clothing.