Finite Difference Discretization Research Articles

A difference finite element method based on nonconforming finite element methods for 3D elliptic problems

In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on P1-element in the z-direction and a finite-element (FE) discretization based on P1NC-nonconforming element in the (x, y)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of P1NC-nonconforming finite element methods in the 2D elliptic equation and the error estimation of the H1-norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the P1 element in the z-direction and the FE method based on the Crouzeix-Raviart element, the P1 linear element, the Park-Sheen element, and the Q1 bilinear element, respectively, in the (x, y)-plane.

Comparing Solutions to the Third-Order Dispersive Partial Differential Equation

In previous decades, many of the practical problems arising in scientific fields such as mathematics, physics, chemistry, biology,and engineering have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the third-order dispersive partial differential equation, has been found to have a plethora of useful applications in different fields such as Newtonian fluid mechanics, optimal control, convection diffusion processes, hydrodynamics, and aerodynamics. A special class of solutions has been studied for the third-order dispersive partial differential equation including exact solutions and approximate solutions. The aim of this article is to compare were the Adomian decomposition method, the lines method, an exponential quartic spline and finite difference discretization method, and the non-polynomial spline methodwith the solution of the third-order dispersive partial differential equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

Densely packed membrane configurations

AbstractWe put forth a simple mathematical model of densely packed fluid membranes and solve for packing configurations that minimize their elastic energy. Numerical calculations are facilitated via a finite-difference discretization scheme. Absent topological constraints, energy-minimizing configurations are found to closely follow solutions of the eikonal equation. These typically involve foliations comprising many closed surfaces. We show how allowing for cuts and creases, with an additional minimization over the total crease energy, generates configurations consisting of a densely packed single sheet.

On the Effect of Liquid Crystal Orientation in the Lipid Layer on Tear Film Thinning and Breakup

AbstractThe human tear film (TF) is thin multilayer fluid film that is critical for clear vision and ocular surface health. Its dynamics are strongly affected by a floating lipid layer and, in health, that layer slows evaporation and helps create a more uniform tear film over the ocular surface. The tear film lipid layer (LL) may have liquid crystalline characteristics and plays important roles in the health of the tear film. Previous models have treated the lipid layer as a Newtonian fluid in extensional flow. We extend previous models to include extensional flow of a thin nematic liquid crystal atop a Newtonian aqueous layer with insoluble surfactant between them. We derive the resulting system of nonlinear partial differential equations for thickness of the LL and aqueous layers, surfactant transport and velocity in the LL. We find that in the limit used here, the liquid crystal director field becomes orientated at a constant angle through the depth of LL. Evaporation is taken into account, and is affected by the LL thickness, internal arrangement of its rod-like molecules, and external conditions. Despite the complexity, this system still represents a significant reduction of the full system. We solve the system numerically via collocation with finite difference discretization in space together with implicit time stepping. We analyze solutions for different internal LL structures and show significant effect of the orientation. Orienting the molecules close to the normal direction to the TF surface results in slower evaporation, and other orientations have an effect on flow, showing that this type of model has promise for predicting TF dynamics.

Optimized five-by-five block preconditioning for efficient GMRES convergence in curvature-based image deblurring

We introduce an enhanced preconditioning technique designed to expedite the convergence of Krylov subspace methods when dealing with non-linear systems of equations featuring a block five-by-five format. This scenario often arises in the context of cell centered finite difference discretizations applied to the mean curvature based image deblurring problem. A thorough spectral analysis of the preconditioned matrices reveals a favorable eigenvalue distribution, leading to accelerated convergence of preconditioned Generalized Minimal Residual (GMRES) methods. Furthermore, we present numerical experiments to showcase the effectiveness of this preconditioner when combined with the flexible GMRES solver for addressing non-linear systems of equations originating from a image deblurring problems. Codes can be obtained at https://github.com/shahbaz1982/Preconditioning.

Efficient pure qP-wave modeling and reverse time migration in tilted transversely isotropic media calculated by a finite-difference approach

Subsurface anisotropy is commonly induced by shale layers, aligned cracks, and fine bedding and has a significant impact on seismic wave propagation. Ignoring anisotropic effects during seismic migration degrades image quality. Therefore, we derive a pure qP-wave equation with high accuracy for modeling seismic wave propagation in tilted transversely isotropic (TTI) media. However, the derived pure qP-wave equation requires a computationally expensive spectral-based method for performing numerical simulations. This is unsuitable for large-scale industrial applications, particularly 3D applications. For numerical efficiency, we first decompose the newly derived wave equation into some fractional differential operators and spatial derivatives. The spatial derivatives can be directly solved by conventional finite-difference (FD) approaches. Then, we use an asymptotic approximation to find an equivalent form of fractional differential operators, obtaining scalar operators that we can discretize with the FD method. Numerical tests show that our TTI pure qP-wave equation with an FD discretization can produce accurate and highly efficient wavefield simulations in TTI media. We also use our TTI pure qP-wave equation with an FD discretization as a forward engine to implement TTI reverse time migration (RTM). Synthetic examples and a field data test demonstrate that our TTI RTM can effectively correct the anisotropic effects, providing high-quality imaging results while maintaining good computational efficiency.

A new higher-order super compact finite difference scheme to study three-dimensional non-Newtonian flows

This work introduces a new higher-order accurate super compact (HOSC) finite difference scheme for solving complex unsteady three-dimensional (3D) non-Newtonian fluid flow problems. As per the author's knowledge, the proposed scheme is the first ever developed higher-order compact finite difference scheme to solve 3D non-Newtonian flow problems. The proposed scheme is fourth-order accurate in space and second-order accurate in time, utilizing only seven adjacent grid points at the (n+1)th time level for the finite difference discretization. A time-marching methodology is employed with pressure calculated via a pressure-correction strategy based on the modified artificial compressibility method. Using the power-law viscosity model, we tackle the benchmark problem of a 3D lid-driven cavity, systematically analyzing the varied rheological behavior of shear-thinning (n = 0.5), shear-thickening (n = 1.5), and Newtonian (n = 1.0) fluids across different Reynolds numbers (Re=1,50,100,200). Both Newtonian and non-Newtonian results are carefully investigated in terms of streamlines, velocity variation, pressure distributions, and viscosity contours, and the computed results are validated with the existing benchmark results. The findings demonstrate excellent agreement with the existing results. It is found that for shear-thinning fluid (n = 0.5), u velocity is higher near the top moving wall than the case of Newtonian (n = 1.0) and shear-thickening fluid (n = 1.5) for all Re values. This extensive analysis, using the new HOSC scheme, not only increases our understanding of non-Newtonian fluid behavior but also provides a robust foundation for future research and practical applications.

Simulation of gas mixture dynamics in a pipeline network using explicit staggered-grid discretization

We develop an explicit staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge boundary values that are more complex than for a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

A robust compact finite difference framework for simulations of compressible turbulent flows

A robust high-order compact finite difference framework is proposed for simulations of compressible turbulent flows with high spectral resolution using a fully collocated variable storage paradigm. Both inviscid and viscous fluxes are assembled at the edge-staggered grid locations. Nonlinear robustness is attained as a consequence of the intrinsic reduction of aliasing errors in the inviscid fluxes due to the spectral behavior of the compact interpolation schemes. Additional robustness is provided by enhancing the spectral resolution of the viscous flux and its divergence at small scales using purely staggered numerical differentiation. Demonstrative simulations have shown numerical stability of the compact finite difference discretization without any type of solution filtering on both Cartesian and curvilinear meshes. For simulations on a curvilinear mesh, a general metric evaluation approach that satisfies the geometric conservation law is proposed. Additional approaches to combining the proposed scheme with approximate Riemann solvers and artificial diffusivities for shock-capturing are also discussed. Along with theoretical analysis, rigorous evaluation and validation of the methodology on canonical tests, including classic two-dimensional simulations, direct numerical simulations, and large-eddy simulations, are used to confirm robustness and accuracy.

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

A symmetric multigrid-preconditioned Krylov subspace solver for Stokes equations

Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of the multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme [1], and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.

Acoustic shape optimization using energy stable curvilinear finite differences

A gradient-based method for shape optimization problems constrained by the acoustic wave equation is presented. The method makes use of high-order accurate finite differences with summation-by-parts properties on multiblock curvilinear grids to discretize in space. Representing the design domain through a coordinate mapping from a reference domain, the design shape is obtained by inverting for the discretized coordinate map. The adjoint state framework is employed to efficiently compute the gradient of the loss functional. Using the summation-by-parts properties of the finite difference discretization, we prove stability and dual consistency for the semi-discrete forward and adjoint problems. Numerical experiments verify the accuracy of the finite difference scheme and demonstrate the capabilities of the shape optimization method on two model problems with real-world relevance.

Re-patterning of cylindrical packing of diblock copolymers under confinement and curvature effects by using approximations of PDE’s involved in the CDS model on polar mesh system

Soft materials, including diblock copolymers, are advancing nanotechnology due to their unique properties, applications materials include energy harvesting, water sanitation, environmental treatment, nanosensors, drug delivery and nanolithography. These materials are light, cheap, efficient, sensitive, durable and more functional, whose new morphologies have been predicted by mathematicians through simulation. This work produces and predicts the pattern of packing of nano-cylinders by using confinement to appreciate the frustration in the packing of nano-cylinders under the influence of curvature. In this contribution, the cell dynamic simulations model is used to examine the impact of circular annular pore confinement on system orientation toward cylindrical morphologies. A 9-point stencil approximates the isotropic Laplacian by finite-difference discretization on a polar grid to meet the requirement of a cell dynamic simulation model. FORTRAN codes are generated for the set of PDEs included in the CDS model. OPEN DX is used to visualise the predicted cylindrical patterns. The consistency of our results with experimental observations makes our research valid and significant.

A novel phase-field lattice Boltzmann framework for diffusion-driven multiphase evaporation

Heat transfer and phase change phenomena, particularly diffusion-driven droplet evaporation, play pivotal roles in various industrial applications and natural processes. Despite advancements in computational fluid dynamics, modeling multiphase flows with large density ratios remains challenging. In this study, we developed a robust and stable conservative Allen–Cahn-based phase-field lattice Boltzmann method to solve the flow field equations. This method is coupled with the finite difference discretization of vapor species transport equation and the energy equation. The coupling between the vapor concentration and temperature field at the interface is modeled by the well-known Clausius–Clapeyron correlation. Our approach is capable of simulations under real physical conditions and is compatible with graphics processing unit architecture, making it ideal for large-scale industrial simulations. Three validation test cases are conducted to demonstrate the consistency of the presented model, including simulations of Stefan flow, the evaporation of suspended droplets containing water, acetone, and ethanol in the air, and the evaporation of a water sessile droplet on a flat surface. The results show that the model is able to predict the behavior and characteristics of each case accurately. Notably, our numerical results exhibit a maximum relative error of approximately 1% in simulations of Stefan flow. In the case of suspended droplet evaporation, the observed maximum difference between the calculated wet bulb temperatures and those derived from psychrometric charts is approximately 0.9 K. Moreover, our analysis of the sessile droplet reveals a good agreement between the results obtained by our model for the evaporative mass flux and those obtained from the existing models in the literature for different contact angles.

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

Numerical solution of convected wave equation in free field using artificial boundary method

AbstractIn this article, we propose two procedures focusing on the computation of the time‐dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430–A1445.), of constructing the Dirichlet‐to‐Dirichlet (DtD)‐type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl‐Glauert‐Lorentz transformation. After that, we obtain the DtD‐type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet‐to‐Neumann‐type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments.

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented.

Random pore model insights into structural and operational parameters for hydrogen-based iron oxide reduction

The hydrogen-based direct reduction of iron oxide (DRI) provides a sustainable approach to cleaner ironmaking. In this research, for analyzing hydrogen-based DRI in pellet scale, the random pore model (RPM), the most sophisticated non-catalytic gas-solid model, is applied to determine the effects of structural parameters, including pore-size parameter (ψ), Thiele modulus (ϕ), and product layer resistance (β), along with operational parameters such as pressure, temperature, and gas composition. The PDE equations of RPM were solved numerically using the method of lines together with a finite-difference discretization in space. The effects of parameters ψ and ϕ were investigated, revealing that the reaction rate increases with larger pores and decreases with higher values of ϕ over length and time. Kinetics resemble the unreacted shrinking core model (USCM) for small particles, while pore structure dominates for large particles. Parameter determination is influenced by ϕ changes, shifting control from chemical kinetics to diffusion, with operating conditions impacting reaction rate enhancement. Investigations revealed that as ϕ increases from 0.454 to 1.454, a 6 % conversion gradient forms from the pellet center to the surface. Additionally, at 700°C, reducing the temperature by 100°C slows conversion by 15.45 minutes while increasing it accelerates conversion by 4.55 minutes.

A comparison of Finite difference schemes to Finite volume scheme for axially symmetric 2D heat equation

In this work, we compare finite difference schemes to finite volume scheme for axially symmetric 2D heat equation with Dirichlet and Neumann boundary conditions. Using cylindrical coordinate geometry, we describe a mathematical model of axially symmetric heat conduction for a stationary, homogeneous isotrophic solid with uniform thermal conductivity in a hollow cylinder with an exact solution in a particular case. We obtain the numerical solution of the PDE adapting finite difference and finite volume discretization techniques. Compared to the exact solution, we explore that the numerical schemes are the sufficient tools for the solution of linear or nonlinear PDE with prescribed boundary conditions. Furthermore, the numerical solution discrepancies in the results obtained from Explicit, Implicit and Crank-Nicolson schemes in Finite Difference Method (FDM) are extremely close to the exact solution in the case of Dirichlet boundary condition. The solution from the Explicit scheme is slightly far from the exactsolution and the solutions from Implicit and Crank-Nicolson schemes are extremely close to the exact solution in the case of Neumann boundary condition. Likewise, the numerical solutions obtained in the Finite volume method (FVM) are extremely close to the exact solution in the case of the Dirichlet boundary condition and slightly away from the exact solution in the case of the Neumann boundary condition.

Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations

We construct a positivity-preserving Lie–Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate 1/4 in time and rate 1/2 in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.