Mixed Finite Volume Research Articles

An upwind-mixed volume element method on changing meshes for compressible miscible displacement problem

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications.

Coupled mixed finite element and finite volume methods for a solid velocity-based model of multidimensional sedimentation

In this paper we introduce and analyze a model of sedimentation based on a solid velocity formulation. A particular feature of the governing equations is given by the fact that the velocity field is non-divergence free. We introduce extra variables such as the pseudostress tensor relating the velocity gradient with the pressure, thus leading to a mixed variational formulation consisting of two systems of equations coupled through their source terms. A result of existence and uniqueness of solutions is shown by means of a fixed-point strategy and the help of the Babuška–Brezzi theory and Banach theorem. Additionally, we employ suitable finite dimensional subspaces to approximate both systems of equations via associated mixed finite element methods. The well-posedness of the resulting coupled scheme is also treated via a fixed-point approach, and hence the discrete version of the existence and uniqueness result is derived analogously to the continuous case. The above is then combined with a finite volume method for the transport equation. Finally, several numerical results illustrating the performance of the proposed model and the full numerical scheme, and confirming the theoretical rates of convergence, are presented.

The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair

The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair

New finite difference unequal-sized Hermite WENO scheme for Navier-Stokes equations

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32] , [33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32] , [33] . Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24] , [25] . So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme.

Compressible Navier–Stokes Equations with Potential Temperature Transport: Stability of the Strong Solution and Numerical Error Estimates

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.

A Mixed Finite Volume Element Method for Time-Fractional Damping Beam Vibration Problem

In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm are analyzed. Numerical examples demonstrate the effectiveness of the algorithm. Finally, the values of different parameter sets are tested, and the test results show that both the damping coefficient and the fractional derivative have significant effects on the model. The results of this paper can be used for the damping modeling of viscoelastic structures.

Research on the thermo-hydraulic response characteristics of pore-fissure media using mixed finite volume method

After lengthy diagenesis and tectonic movement, a rock mass inevitably develops many pores and micro-fissures. A numerical simulation method was employed to study the thermal response characteristics of the rock mass under temperature seepage coupling by treating it as a pore-fissure medium and considering its anisotropic properties. Based on the mixed finite volume method (FVM), a numerical scheme of the governing equation for the temperature seepage coupling of the pore-fissure medium is derived, with the program solution module independently written in C++. On this basis, a numerical test model of the fissured mudstone is established to analyze the distribution of the rock mass temperature field under various thermal conductivities and the influence of fissure permeability on the seepage field. The mixed FVM results revealed that the temperature and water pressure distributions near the fissure were closely related to the directionality of thermal conductivity in the rock mass, as well as the thermal conductivity and permeability coefficient, respectively, of the fissure itself. Comparison with results from the finite element software ABAQUS demonstrated significant advantages of the proposed method when solving temperature and seepage problems in discontinuous geological bodies containing hiatuses, mutations, and fissures.

Existence of Dissipative Solutions to the Compressible Navier-Stokes System with Potential Temperature Transport

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a strong solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.

Law of Nuclide Migration in Clayey Rocks considering Diffusion and Fluid Transport

A core concern in the research on deep geological disposal of high-level radioactive waste is the migration of radionuclides in geological bodies. Most studies on radionuclide migration consider the role of only the rock fissures without incorporating the influence of the rock matrix. In this paper, the rock mass for geological disposal of high-level radioactive waste is regarded as a fissure-pore medium. Considering the influences of radionuclide diffusion and fluid transport on radionuclide migration in the process of disposal, the governing equation of radionuclide migration and evolution in the pore-fissure medium is established. The numerical scheme of the governing equation is given based on the mixed finite volume method (FVM), using our program solution module written in C++. On this basis, the numerical test model with fissures was developed, which analyzed the radionuclide migration law in clayey rocks under various fissure and rock matrix diffusion coefficients and hydraulic conductivities. The simulation results are compared with finite element method results, revealing the superiority of the mixed FVM method in solving problems of radionuclide migration in discontinuous geological bodies containing hiatuses, mutations, and fissures. The study provides a theoretical basis for evaluating the safety, feasibility, and suitability of geological disposal repositories for high-level radioactive waste in terms of radionuclide migration.

“Mixed” Meshless Time-Domain Adaptive Algorithm for Solving Elasto-Dynamics Equations

A time-domain adaptive algorithm was developed for solving elasto-dynamics problems through a mixed meshless local Petrov-Galerkin finite volume method (MLPG5). In this time-adaptive algorithm, each time-dependent variable is interpolated by a time series function of n-order, which is determined by a criterion in each step. The high-order series of expanded variables bring high accuracy in the time domain, especially for the elasto-dynamic equations, which are second-order PDE in the time domain. In the present mixed MLPG5 dynamic formulation, the strains are interpolated independently, as are displacements in the local weak form, which eliminates the expensive differential of the shape function. In the traditional MLPG5, both shape function and its derivative for each node need to be calculated. By taking the Heaviside function as the test function, the local domain integration of stiffness matrix is avoided. Several numerical examples, including the comparison of our method, the MLPG5–Newmark method and FEM (ANSYS) are given to demonstrate the advantages of the presented method: (1) a large time step can be used in solving a elasto-dynamics problem; (2) computational efficiency and accuracy are improved in both space and time; (3) smaller support sizes can be used in the mixed MLPG5.

A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed.

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW L-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.

Numerical simulation of spherical bubble collapse by a uniform bubble pressure approximation and detailed description of heat and mass transfer with phase transition

A mathematical model for the simulation of the dynamics of spherical vapor-air bubbles and its numerical implementation is presented. Heat and mass transfer and phase transition in terms of evaporation and condensation as well as air absorption and desorption are considered. Flow variables are discretized by a mixed finite volume / finite difference scheme and solved either by a Crank-Nicolson or Runge-Kutta scheme. Due to the assumption of homogeneous bubble pressure (homobaricity), the solution of momentum equations is restricted to the Rayleigh-Plesset equation which makes the model computationally efficient. The model is validated by measurement data for bubble growth and applied to bubble collapse and rebound. By a comparison with Navier-Stokes results from literature, the homobaricity assumption is shown to be appropriate even in the last stage of bubble collapse. The relation of the local velocity and temperature field with heat and mass transfer is discussed. Equilibrium bubble interface conditions (liquid and gaseous side have the same temperature and chemical potential) are compared to non-equilibrium conditions and are shown to yield the same local velocity and temperature field for each stage of bubble collapse and rebound.

Consistency, convergence and error estimates for a mixed finite element–finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data

Abstract We study convergence of a mixed finite element-finite volume scheme for the compressible Navier–Stokes equations in the isentropic regime under the full range $1<\gamma <\infty $ of the adiabatic coefficients $\gamma $ for the problem with general nonzero inflow–outflow boundary conditions. We propose a modification of Karper’s scheme (2013, A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math., 125, 441–510) in order to accommodate the nonzero boundary data, prove existence of its solutions, establish the stability and uniform estimates, derive a convenient consistency formulation of the balance laws and use it to show the weak convergence of the numerical solutions to a dissipative solution with the Reynolds defect introduced in Abbatiello et al. (2021, Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst., 41, 1--28). If the target system admits a strong solution then the convergence is strong towards the strong solution. Moreover, we establish the convergence rate of the strong convergence in terms of the size of the space discretization $h$ (which is supposed to be comparable with the time step $\varDelta t$). In the case of the nonzero inflow–outflow boundary data all results are new. The latter result is new also for the no-slip boundary conditions and adiabatic coefficients $\gamma $ less than the threshold $3/2$.

Mixed Finite Volume Element Method for Vibration Equations of Beam with Structural Damping

In this paper, for the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed. Numerical examples are provided to confirm the theoretical results. In the end, we test the value of δ to observe its influence on the model.

A Near-Shore Linear Wave Model with the Mixed Finite Volume and Finite Difference Unstructured Mesh Method

The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models have been developed to predict the wave and current state and their interactions. Most models, however, have relied on the two-model approach in which the wave model is developed independently of the current model and the two are coupled together through a separate steering module. In this study, a new wave model is developed and embedded in an existing two-dimensional (2D) depth-integrated current model, SRH-2D. The work leads to a new wave–current model based on the one-model approach. The physical processes of the new wave model are based on the latest third-generation formulation in which the spectral wave action balance equation is solved so that the spectrum shape is not pre-imposed and the non-linear effects are not parameterized. New contributions of the present study lie primarily in the numerical method adopted, which include: (a) a new operator-splitting method that allows an implicit solution of the wave action equation in the geographical space; (b) mixed finite volume and finite difference method; (c) unstructured polygonal mesh in the geographical space; and (d) a single mesh for both the wave and current models that paves the way for the use of the one-model approach. An advantage of the present model is that the propagation of waves from deep water to shallow water in near-shore and the interaction between waves and river inflows may be carried out seamlessly. Tedious interpolations and the so-called multi-model steering operation adopted by many existing models are avoided. As a result, the underlying interpolation errors and information loss due to matching between two meshes are avoided, leading to an increased computational efficiency and accuracy. The new wave model is developed and verified using a number of cases. The verified near-shore wave processes include wave shoaling, refraction, wave breaking and diffraction. The predicted model results compare well with the analytical solution or measured data for all cases.

Coupled compositional flow and geomechanics modeling of fractured shale oil reservoir with confined phase behavior

Shale oil flow is commonly affected by the confined phase behavior, complex fracture geometries, and geomechanical effect, which are not fully considered in the conventional compositional simulators. In this study, a coupled compositional flow and geomechanics model is presented to incorporate these effects for more accurate prediction on shale oil production performance. The compositional model considering the confined phase behavior is used to model the multiphase flow in shale formations, in which the capillary pressure is introduced into the workflows of phase stability test and flash calculation. The embedded discrete fracture model (EDFM) is adopted to explicitly describe the complex fracture geometries. The deformations of hydraulic fractures and natural fractures are handled by different constitutive models. A mixed finite-volume and finite-element (FVM-FEM) scheme is used to discretize the flow and geomechanics equations, and the coupled problem is iteratively solved by the fixed-stress splitting method. Based on the proposed model, simulations are conducted on a multi-stage fractured shale reservoir to evaluate the effects of confined phase behavior, hydraulic fracture deformation, and natural fracture network on the production performance. Results show that the bubble point pressure is suppressed by the confined phase behavior effect, which leads to higher oil production. The hydraulic fracture deformation takes effect only when the hydraulic fractures cannot be treated as infinite-conductivity under poorly-propped condition. The oil production is greatly promoted with the increase of natural fracture density due to the increase of interface area between matrix and fractures.

Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimal a priori error estimates for the unknown and auxiliary variables in the L2Ω norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.

Verification of Anisotropic Mesh Adaptation for Turbulent Simulations over ONERA M6 Wing

Unstructured anisotropic mesh adaptation is known to be an efficient way to control discretization errors in computational fluid dynamics simulations. Method verification is required to provide the confidence for routine use in production analysis. The current work aims at verification of anisotropic mesh adaptation for Reynolds-averaged Navier–Stokes simulations over the ONERA M6 wing. The present verification study is performed using four different flow solvers, three different implementations of the metric field, and three mesh mechanics packages. Two of the flow solvers use stabilized finite element discretizations (FUN3D-SFE and general geometry Navier–Stokes), one uses finite volume discretization (FUN3D-FV), and the last one uses mixed finite volume and finite element discretizations (Wolf). The mesh adaptation is based on an error estimator that aims to control the quadratic error term in the linear interpolation of the Mach number. Two sets of adaptations were performed; the first one controls the interpolation error in the norm, and the second one controls the interpolation error in the norm. Convergence studies were performed on the forces and the pitching moment using all four solvers, and the results are compared with previously verified convergence studies on fixed (nonadapted) meshes. Both the forces and pitching moment on adapted meshes are found to be converging to the fine-mesh values faster than those on fixed meshes. In addition to forces and moments, the convergence of surface pressure and skin-friction coefficients at various measurement locations on the wing are also presented. Adapted-mesh surface pressure distributions agree with the fine fixed mesh pressure distributions. Adapted-mesh skin-friction distributions contain high-frequency noise with mean values approaching the fixed mesh pressure skin-friction distributions.