Cell-centered Finite Difference Research Articles

A cell-centered implicit finite difference scheme to study wave propagation in acoustic media: A numerical modeling

In the present paper, we present a Cell-Centered Implicit Finite Difference (CCIFD) operator-based numerical scheme for the propagation of acoustic waves that is very effective, accurate, and small in size. This scheme requires fewer estimation points than the traditional central difference derivative operator. Any numerical simulation is significantly impacted by the precision of a numerical derivative. Long stencils can deliver excellent accuracy while also minimizing numerical anisotropy error. However, a long stencil requires a lot of computational resources, and as these derivatives get bigger, they could start to look physically unrealistic due to contributions from nodes located extremely far, wherein the derivative is local in nature. Furthermore, using such lengthy stencils at boundary nodes may result in errors. The present article investigates a cell-centered fourth order finite difference scheme to model acoustic wave propagation which utilizes a lesser number of nodes in comparison to the traditional Central Difference (CD) operator. However, in general the implicit derivative operator has high computational cost and therefore despite its significant advantages it is generally avoided to be implemented in applications. This serves as a motivation for the present paper to explore a technique called CCIFD that significantly decreases the computational expense by nearly fifty percent. Additionally, spectral characterization of the CCIFD derivative operator has been analyzed and discussed. Finally, the wave propagation has been numerically simulated in 2-dimensional homogeneous and Marmousi model using CCIFD scheme to validate the applicability and stability of the scheme.

A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

In this paper, we propose an efficient numerical method for a comprehensive infection model that is formulated by a system of nonlinear coupling advection-diffusion-reaction equations. Using some subtle mixed explicit-implicit treatments, we construct a linearized and decoupled discrete scheme. Moreover, the proposed scheme is capable of preserving the positivity of variables, which is an essential requirement of the model under consideration. The proposed scheme uses the cell-centered finite difference method for the spatial discretization, and thus, it is easy to implement. The diffusion terms are treated implicitly to improve the robustness of the scheme. A semi-implicit upwind approach is proposed to discretize the advection terms, and a distinctive feature of the resulting scheme is to preserve the positivity of variables without any restriction on the spatial mesh size and time step size. We rigorously prove the unique existence of discrete solutions and positivity-preserving property of the proposed scheme without requirements for the mesh size and time step size. It is worthwhile to note that these properties are proved using the discrete variational principles rather than the conventional approaches of matrix analysis. Numerical results are also provided to assess the performance of the proposed scheme.

An efficient bound-preserving and energy stable algorithm for compressible gas flow in porous media

Due to the great significance of the natural gas and shale gas, it is becoming increasingly important to simulate compressible gas flow in porous media. The model obeys an energy dissipation law as well as molar density must be positive and bounded in terms of the equation of state. For the purpose of eliminating nonphysical solutions as well as improving the stability in practical simulation, preservation of these properties is essential for a promising numerical method, but it is actually challenging due to the strong nonlinearity and complexity of the model. In this paper, we propose an efficient linearized numerical scheme that inherits the energy dissipation law as well as preserves the boundedness of molar density. Specifically, to treat the logarithmic type Helmholtz free energy density determined by the Peng-Robinson equation of state, we propose a novel adaptive stabilization approach involving the second derivatives of the convex energy terms. At each time step, the stabilization parameter is adaptively updated by a simple and explicit formula to ensure the energy dissipation law. The stabilized and linearized chemical potential allows to formulate the local mass conservation equation as an equivalent convection-diffusion form, and from this, an adaptive time step strategy is proposed to preserve the positivity and boundedness of molar density. The calculation of the time step size is fully explicit and easy to implement. Additionally, the fully discrete scheme is constructed using the conservative cell-centered finite difference method with the upwind strategy, and thus, it enjoys the local mass conservation. Numerical results are also presented to demonstrate the excellent performance of the proposed scheme.

Energy Stable and Mass Conservative Numerical Method for Gas Flow in Porous Media with Rock Compressibility

Taking rock compressibility into consideration brings up a quite challenging issue to modeling and numerical simulation of gas flow in porous media. To develop efficient numerical methods for such a problem, on the basis of the existing model, we, for the first time, propose an alternative formulation by introducing a free energy accounting for rock compressibility and by using chemical potential gradient instead of pressure gradient as the primary driving force. This formulation has a distinct feature that it is proved to satisfy an energy dissipation law. We further propose an efficient energy stable time discrete scheme using the improved energy factorization approach to tackle the Helmholtz free energy density determined by the Peng--Robinson equation of state. The discrete pressure is taken carefully using the discrete chemical potential and free energy such that the discrete energy dissipation law can be preserved. The fully discrete scheme is constructed based on the cell-centered finite difference method with the upwind strategy, which features full mass conservation and positivity preservation of molar density under proper conditions. Numerical results are also provided to show the performance of the proposed scheme and validate theoretical analysis.

Heterogeneous Stefan problem and permafrost models with P0-P0 finite elements and fully implicit monolithic solver

<abstract><p>We consider heat conduction models with phase change in heterogeneous materials. We are motivated by important applications including heat conduction in permafrost, phase change materials (PCM), and human tissue. We focus on the mathematical and computational challenges associated with the nonlinear and discontinuous character of constitutive relationships related to the presence of free boundaries and material interfaces. We propose a monolithic discretization framework based on lowest order mixed finite elements on rectangular grids well known for its conservative properties. We implement this scheme which we call P0-P0 as cell centered finite differences, and combine with a fully implicit time stepping scheme. We show that our algorithm is robust and compares well to piecewise linear approaches. While various basic theoretical properties of the algorithms are well known, we prove several results for the new heterogeneous framework, and point out challenges and open questions; these include the approximability of fluxes by piecewise continuous linears, while the true flux features a jump. We simulate a variety of scenarios of interest.</p></abstract>

An energy stable linear numerical method for thermodynamically consistent modeling of two-phase incompressible flow in porous media

In this paper, we consider numerical approximation of a thermodynamically consistent model of two-phase flow in porous media, which obeys an intrinsic energy dissipation law. The model under consideration is newly-developed, so there is no energy stable numerical scheme proposed for it at present. This model consists of two nonlinear degenerate parabolic equations and a saturation constraint, but lacking an independent equation for the pressure, so for the purpose of designing efficient numerical scheme, we reformulate the model forms as well as the free energy function, and further prove the corresponding energy dissipation inequality. Based on the alternative reformulations, using the invariant energy quadratization approach and subtle semi-implicit treatments for the pressure and saturation, we for the first time propose a linear and energy stable numerical method for this model. The fully discrete scheme is devised combining the upwind approach for the phase mobilities and the cell-centered finite difference method. The unique solvability of numerical solutions and unconditional energy stability are rigorously proved for both the semi-discrete time scheme and the fully discrete scheme. Moreover, the scheme can guarantee the local mass conservation for both phases. We also show that the upwind mobility approach plays an essential role in preserving energy stability of the fully discrete scheme. Numerical results are presented to demonstrate the performance of the proposed scheme.

Analysis of the CCFD method for MC-based image denoising problems

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method

High-order compact interpolation schemes appropriate for multiscale flows are studied within a cell-centered finite difference method (CCFDM) framework where the robustness of high-order schemes on curvilinear grids can be greatly enhanced due to the satisfaction of geometric conservation law. Two types of compact interpolations are mainly developed in this paper for shock-free flows and shock-embedded flows respectively. The present compact schemes are verified to be superior over the explicit counterparts with same orders in terms of the spectral characteristics. Regarding the shock-free flows, low-dissipation low-dispersion properties are achieved by the spectral optimization. Three optimized compact schemes (Opt4, Opt6 and Opt8) are further validated to be attractive for shock-free problems by carrying out benchmarks from computational aeroacoustics workshops and two typical turbulence cases: Tayler–Green vortex and decaying isotropic turbulence. Regarding high-speed flows in the presence of shock waves, the shock-capturing capability is realized by extending the weighting technique to the compact interpolations. The criteria to choose optimally compact nonlinear sub-stencils on a most general compact global stencil are presented. Interestingly, the explicit WENO-type schemes can be reverted within the proposed compact framework. Three nonlinear compact schemes (UI5, CI6 and CI8) on two practical stencils are analyzed and further compared with their explicit counterparts by a series of numerical experiments. The compact ones are superior to explicit ones in resolving rich flow structures as well as discontinuities.

Investigation on rod-airfoil noise with high-order cell-centered finite difference method and acoustic analogy

This paper focuses on the radiation noise of a rod-airfoil configuration based on a high-order cell-centered finite difference method (CCFDM) and Ffowcs Williams-Hawkings (FW-H) acoustic analogy. To reduce numerical dissipation and dispersion, a class of optimized compact schemes is firstly proposed and then validated with benchmark cases from computational aeroacoustic workshops. For turbulence, the delayed detached eddy simulation (DDES) with Spalart-Allmaras model is adopted and is validated through a canonical periodic hill problem. For the far-field radiated acoustics, the FW-H acoustic analogy is utilized with validation of laminar flow past NACA0012. Finally, a rod-airfoil configuration is investigated. The results show reasonable agreement with the experimental data in terms of both near-field aerodynamics and far-field acoustics. The interaction between the upstream turbulence wake from the rod and the downstream airfoil is evidenced to be the dominant contributor to the far-field noise.

A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

The Peng--Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and the petroleum industry due to its excellent accuracy in predicting the thermodynamic properties of a wide variety of materials, especially hydrocarbons. Although great effort has been made to construct efficient numerical methods for the diffuse interface models with PR-EoS, there is still not a linear numerical scheme that can be proved to preserve the original energy dissipation law. In order to pursue such a numerical scheme, we propose a novel energy factorization (EF) approach, which first factorizes an energy function into a product of several factors and then treats the factors using their properties to obtain the semi-implicit linear schemes. We apply the EF approach to deal with the Helmholtz free energy density determined by PR-EoS, and then propose a linear semi-implicit numerical scheme that inherits the original energy dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum principle in both time semidiscrete form and cell-centered finite difference fully discrete form under certain conditions. Numerical results are presented to demonstrate the stability and efficiency of the proposed scheme.

A locally and globally phase-wise mass conservative numerical algorithm for the two-phase immiscible flow problems in porous media

In this work, we introduce a novel numerical method to solve the problem of two-phase immiscible flow in porous media that is conservative to both phases. In the widely used implicit pressure, explicit saturation (IMPES) scheme, the conservation of mass of both the two phases are summed to form an equation involving the total Darcy’s velocity. In the discretization of such an equation it becomes difficult to enforce the conservation of mass of each phase. To guarantee the conservation of mass of both phases locally and hence globally, we introduce a scheme in which the time discretization of the mass conservation equations is considered separately. Cell-centered finite difference (CCFD) methods are adopted for spatial discretization, where the variables of fluid properties (i.e. relative permeability and mobility) are upwinded separately according to the velocity of each phase and not according to the total velocity. Furthermore, this new scheme updates all phase velocities and uses them to update the corresponding phase saturation. In addition, a two-scale of time-splitting methods are adopted for pressure equation and saturation equations to improve the computational efficiency. For the sake of simplicity, we show a number of examples of two-phase system in two-dimensional geometry solved using the new scheme. It is shown that the new scheme is more embracing the physics and it can be more accurate than traditional IMPES scheme, particularly for the cases in which the phase velocities are in opposite direction, and conventional IMPES schemes fail.

High-order adapter schemes for cell-centered finite difference method

The present paper proposes that reconstruction scheme and interpolation scheme can be converted into each other through two series of adapter schemes, which include reconstruction-to-interpolation (RI) adapter schemes and interpolation-to-reconstruction (IR) adapter schemes. For the high-order spatial discretization of the compressible Navier-Stokes equations, the RI adapter schemes can be used to derive interpolation schemes for the interpolation-based cell-centered finite difference method from the available optimized reconstruction schemes. The main advantage of the interpolation-based cell-centered finite difference method is the capability to realize high-order discretization on curvilinear grids with both shock-capturing capability and satisfaction of the geometric conservation law. In the present paper, we first derive the IR adapter schemes by comparing the difference schemes with their strong conservative forms. We then develop the corresponding RI adapter schemes by inversing the IR adapter schemes. Thereafter, the applications to the one-dimensional linear wave equation and the one-dimensional inviscid Burgers' equation have been briefly discussed. Finally, to demonstrate the application to three-dimensional Navier-Stokes equations, three highly optimized nonlinear reconstruction schemes are adapted into the corresponding interpolation ones through RI adapter schemes, which include WENO-CU6, WGVC-WENO7 and OMP6 schemes. The new interpolation schemes from adapters are compared with their original reconstruction ones through several benchmark cases. No noticeable robustness loss or accuracy loss has been found in these cases, indicating the effectiveness of the adapter schemes. No obvious increase in time cost has been observed, indicating the efficiency of the adapter schemes.

Numerical analysis of one dimensional motion of magma without mass forces

In this paper, we present numerical treatment for a compaction-driven Darcian flow viscoelastic rock magma model. This problem is a strongly coupled system of one quasi-linear parabolic equation and one integro-differential equation for the density and the porosity of the flow. The numerical discretization uses cell-centered finite difference method, combined with semi-implicit and implicit time stepping. Implicit–explicit schemes, as well as implicit–explicit iterative algorithms have been developed to solve the corresponding discrete problems. Some properties (positivity, boundness, conservation) of the numerical solutions are investigated. Convergence study of the iteration processes is also presented. The efficiency and the accuracy of the proposed methods are illustrated numerically by test examples with near-real data.

Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media

In the current paper, an iterative algorithm is developed to simulate the problem of two-phase flow with heat transfer in porous media. The convective body force caused by heat transfer is described by Boussinesq approximation throughout with the governing equations, namely, pressure, saturation, and energy. The two coupled equations of pressure and saturation are solved using the implicit pressure-explicit saturation (IMPES) scheme, while the energy equation is treated implicitly, and the scheme is called iterative implicit pressure, explicit saturation, implicit temperature (I-IMPES-IMT). In order to calculate the pressure implicitly, the equations of pressure and saturation are coupled by linearizing the capillary pressure which is a function of saturation. After that, the equation of saturation is solved explicitly. Then, the velocity is computed which is used in the energy equation to calculate the temperature implicitly. The cell-centered finite difference (CCFD) method is utilized for spatial discretization. Furthermore, a relaxation factor along is used with the Courant–Friedrichs–Lewy (CFL) condition. Finally, in order to illustrate the efficiency of the developed algorithm, error estimates for saturation and temperature for different values of time steps and number of iterations are presented. Moreover, numerical examples of different physical scenarios of heterogamous media are presented.

Numerical Study of the Magnetic Field Effect on Ferromagnetic Fluid Flow and Heat Transfer in a Square Porous Cavity

A numerical study of ferromagnetic-fluid flow and heat transfer in a square porous cavity under the effect of a magnetic field is presented. The water-magnetic particle suspension is treated as a miscible mixture and, thus, the magnetization, density and viscosity of the ferrofluid are obtained. The governing partial-differential equations were solved numerically using the cell-centered finite-difference method for the spatial discretization, while the multiscale time-splitting implicit method was developed to treat the temporal discretization. The Courant–Friedrichs–Lewy stability condition (CFL < 1) was used to make the scheme adaptive by dividing time steps as needed. Two cases corresponding to Dirichlet and Neumann boundary conditions were considered. The efficiency of the developed algorithm as well as some physical results such as temperature, concentration, and pressure; and the local Nusselt and Sherwood numbers at the cavity walls are presented and discussed. It was noticed that the particle concentration and local heat/mass transfer rate are related to the magnetic field strength, and both pressure and velocity increase as the strength of the magnetic was increased.

A fully implicit, conservative, non-linear, electromagnetic hybrid particle-ion/fluid-electron algorithm

The quasi-neutral hybrid model with kinetic ions and fluid electrons is a promising approach for bridging the inherent multi-scale nature of many problems in space and laboratory plasmas. Here, a novel, implicit, particle-in-cell based scheme for the hybrid model is derived for fully 3D electromagnetic problems with multiple ion species, which features global mass, momentum and energy conservation. The scheme includes sub-cycling and orbit-averaging for the ions, with cell-centered finite differences and implicit midpoint time advance. To reduce discrete particle noise, the scheme allows arbitrary-order shape functions for the particle-mesh interpolations and the application of conservative binomial smoothing. The algorithm is verified for a number of test problems to demonstrate the correctness of the implementation, the unique conservation properties, and the favorable stability properties of the new scheme. In particular, there is no indication of unstable growth of the finite-grid instability for a population of cold ions drifting through a uniform spatial mesh, in a set-up where several commonly used non-conservative schemes are highly unstable.

Optimized low-dissipation and low-dispersion schemes for compressible flows

Although interpolation-based finite difference methods have shown great success in the simulation of complex configurations, the schemes currently are still inappropriate in many circumstances due to their poor spectral properties. In this paper, a class of optimized low-dissipation and low-dispersion schemes is developed with the cell-centered finite difference method (Liao, J. Comput. Phys., 284:419–433, 2015). Firstly, the linear schemes with two free parameters which dominate the dispersion and dissipation properties independently are derived. Secondly, the two free parameters are optimized separately according to specific criteria to obtain the optimized linear schemes. Thirdly, for shock-capturing capability, the optimized non-linear schemes are acquired from the optimized linear schemes using the multi-stencil weighting technique. Meanwhile, a new series of weighting functions and smoothness indicators are derived following the idea of Hu (Hu, J. Comput. Phys., 229:8952–8965, 2010) for the optimized non-linear schemes. Finally, several benchmark cases are validated with the developed optimized linear and non-linear schemes. In flows with shocks, the optimized non-linear schemes can give more flow details than the non-linear reference schemes without noticeable robustness loss. In the low-speed problems, the optimized linear schemes perform much better than the corresponding upwind linear schemes. Additionally, satisfactory results are observed on the wavy grid, indicating the optimized schemes with cell-centered finite difference method are promising.

A new central compact finite difference scheme with high spectral resolution for acoustic wave equation

Based on the existing cell-node and cell-centered compact finite difference schemes, we developed a new central compact scheme with a high spectral resolution for the acoustic wave equation. In the new scheme, both the function values on the cell-nodes and cell-centers are used to compute the second-order spatial derivatives on the cell-nodes. The cell-centered values are stored and updated as independent variables in the modeling. The spatial derivatives on the cell-centers are evaluated by half shifting the indices in the formula designed for the cell-nodes. Compared to the conventional compact interpolation scheme, the proposed approach can avoid introducing transfer errors. Either Taylor-series expansion-based or optimized least-squares-based methods are used to calculate the finite difference coefficients. Theoretical analysis and synthetic examples demonstrate that the optimized least-squares-based method can provide higher accuracy than the Taylor-series expansion-based method. This new scheme is not a simple combination of the cell-node and cell-centered compact schemes and outperforms them in three scenarios. Firstly, it can promise higher accuracy considering the same formal truncation errors and model parameters. Thus, it can maintain superior precision while using a shorter spatial finite difference stencil. Secondly, compared to the cell-node compact scheme with half of grid spacing, the new scheme can yield equally as accurate results with less time consuming, together with saving approximately 25% and 29% of memory in 2D and 3D modeling, respectively. Finally, for similar memory requirements, the new method can more efficiently provide solutions with higher accuracy. The synthetic examples on the 2D homogeneous and the 3D horizontally-layered models demonstrate the advantages of the proposed scheme. The numerical simulations with 2D Marmousi model further validate its accuracy, efficiency and flexibility in complex media.

Overlapping localized exponential time differencing methods for diffusion problems

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.

A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids

We present a mass-conservative full approximation storage (FAS) multigrid solver for cell-centered finite difference methods on block-structured, locally cartesian grids. The algorithm is essentially a standard adaptive FAS (AFAS) scheme, but with a simple modification that comes in the form of a mass-conservative correction to the coarse-level force. This correction is facilitated by the creation of a zombie variable, analogous to a ghost variable, but defined on the coarse grid and lying under the fine grid refinement patch. We show that a number of different types of fine-level ghost cell interpolation strategies could be used in our framework, including low-order linear interpolation. In our approach, the smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions at the coarse-fine interface. To maintain global mass conservation, we need only modify the usual FAS algorithm by correcting the coarse-level force function at points adjacent to the coarse-fine interface. We demonstrate through simulations that the solver converges geometrically, at a rate that is h-independent, and we show the generality of the solver, applying it to several nonlinear, time-dependent, and multi-dimensional problems. In several tests, we show that second-order asymptotic (h→0) convergence is observed for the discretizations, provided that (1) at least linear interpolation of the ghost variables is employed, and (2) the mass conservation corrections are applied to the coarse-level force term.