Order Accurate Finite Difference Scheme Research Articles

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

Effects of Variable Thermal Conductivity and Grashof Number on Non-Darcian Natural Convection Flow of Viscoelastic Fluids with Non Linear Radiation and Dissipations

Effects of variable thermal conductivity, porosity and Grashof number on natural convection of viscoelastic fluid flow and heat transfer are studied and discussed. The nonlinear radiation and dissipations are taken into consideration. The fluid flows through non-Darcy porous medium which lies between two heated vertical plates that are kept at constant, but different, temperatures. High order accurate finite difference schemes are introduced to solve the governing equations. The coupled nonlinear differential equations are linearized and iterations are used to approximate that linearized terms. The finite difference method transforms the coupled linearized differential (momentum and energy) equations to a linear system of algebraic equations. An error analysis is introduced by refinement of mesh size and comparisons with available previous results. Effects of fluid and heat parameters on the velocity field, temperature, skin friction factor and Nusselt number are tabulated and plotted. The present results and comparisons show that the numerical solution is of excellent agreement with previous analytical and numerical solution.

PittPack: An open-source Poisson’s equation solver for extreme-scale computing with accelerators

We present a parallel implementation of a direct solver for Poisson’s equation on extreme-scale supercomputers with accelerators. We introduce a chunked-pencil decomposition as the domain-decomposition strategy to distribute work among processing elements to achieve improved scalability at high counts of accelerators. Chunked-pencil decomposition enables overlapping MPI communication and data transfer between the central processing units (CPUs) and the graphics processing units (GPUs). It enables contiguous message transfer among the nodes and improves data locality by keeping neighboring elements in adjacent memory locations while permitting the use of shared memory for certain segments of the algorithm when possible. We study two different communication patterns within the chunked-pencil decomposition. The first pattern fully overlaps the communication with data transfer and aims to speedup the overall turnaround time. The second pattern concentrates on low memory usage and is more network friendly than the first pattern for computations at extreme scale. In our parallel implementation, we interleave OpenACC with MPI to support computations on the GPU or the CPU. The numerical solution and its formal second order of accuracy is verified using the method of manufactured solutions for various combinations of boundary conditions. Additionally, we used PittPack within an incompressible flow solver to further validate its accuracy and as well as demonstrate its versatility as a software package. We performed weak scaling analysis with up to 1.1 trillion Cartesian mesh points distributed over 16384 GPUs on a petascale leadership class supercomputer. Program SummaryProgram Title: PittPackProgram Files doi:http://dx.doi.org/10.17632/59ktwdby4r.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: An elliptic Poisson’s equation for pressure arises in the numerical solution of incompressible fluid flow. This equation needs to be solved accurately to enforce conservation of mass principle.Solution method: The Poisson’s equation is discretized with a second order accurate finite difference scheme. The resulting system of linear equations is solved with a direct method that employs fast Fourier transforms in the horizontal plane and a tridiagonal matrix solver in the third direction.Additional comments including restrictions and unusual features: The mesh has to be directionally uniform. Due to adoption of FFT in the horizontal plane, the boundary conditions in each direction must be of the same type. Each direction in the horizontal plane has to be partitioned by an equal integer number such that n2 parallel processes are deployed at run-time.

Construction of invariant compact finite-difference schemes.

In this paper we propose a method, which is based on equivariant moving frames, for development of high-order accurate invariant compact finite-difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of partial differential equations (PDEs) are used to transform independent and dependent variables and derivative terms of compact finite-difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry-preserving) compact numerical schemes is found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite-difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite-difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in one or two dimensions, the inviscid Burgers' equation in one dimension, viscous Burgers' equation in one or two dimensions, spherical Burgers' equation in one dimension, and shallow water equations in two dimensions). Results obtained from our numerical simulations indicate that invariant compact finite-difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, noninvariant base numerical schemes considered here.

A New High Order Accurate, Finite Difference Method on Quasi-variable Meshes for the Numerical Solution of Three Dimensional Poisson Equation

In this paper, a compact high order accurate finite difference scheme on non-uniform meshes network to solve the 3D Poisson equation is presented. We considered here the development of classical Numerov’s method in combination with nonuniform quasi-variable meshes network. The theoretical validation of the present scheme is considered by means of irreducibility and monotonicity criteria of the coefficient matrix obtained from difference equations. Numerical tests are presented to validate the theoretical prediction of the proposed scheme.

Compact High Order Accurate Schemes for the Three Dimensional Wave Equation

We construct a family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed. High order accuracy is of key importance for the numerical simulation of waves as it reduces the dispersion error (i.e., the pollution effect). The schemes that we propose are built on a stencil that has only three nodes in any coordinate direction or in time, which eliminates the need for auxiliary initial or boundary conditions. These schemes are implicit in time and conditionally stable. A particular scheme with the maximum Courant number can be chosen within the proposed class. The inversion at the upper time level is done by FFT for constant coefficients and multigrid for variable coefficients, which keeps the overall complexity of time marching comparable to that of a typical explicit scheme.

A transformation-free ψ-v formulation of the Navier–Stokes equations on compact nonuniform grids

In this work, we have developed a second order accurate compact finite difference scheme for the biharmonic form of the steady-state Navier–Stokes (N–S) equations on nonuniform Cartesian grids without transformation. Contrary to the schemes developed on rectangular Cartesian grids for N–S equations that could handle only rectangular boundaries, the proposed scheme can easily accommodate bluff bodies with curved boundary. The scheme is applied to one problem having analytical solution and three fluid flow problems having different complexities. While one of them, viz., the lid-driven cavity depict internal flows, the flow past a circular cylinder and a flat plate immersed in uniform flow represents external flows. The scheme is shown to handle both Dirichlet and Neumann boundary conditions with equal ease. The efficiency of the scheme is demonstrated not only by its ability to capture smaller scales up to post quaternary level of vortices for the flow in lid-driven cavity, but also its robustness in handling non-rectangular boundaries of immersed bodies. While the scheme has handled the boundary of an immersed body not necessarily aligned to the grid lines for one of the problems, for another, it has tackled a curved boundary on Cartesian grid without the need of roping in any immersed interface in the process. Its efficiency is further asserted by the proximity of the computed solutions to available experimental results.

Effect of cladding on thermal behavior of nuclear fuel element with non-uniform heat generation

In nuclear power reactors if the maximum temperature (2000ᵒC) of the fuel element like Uranium-dioxide increments and reaches 2865ᵒC then the fuel element melts down leading to heavy loss and damage. To maintain this maximum temperature well below the allowable limit the heat in the fuel element should be conducted and dissipated to the surrounding medium. In this article, the numerical prediction and comparison of thermal performance characteristics of a nuclear fuel element with and without cladding (bare fuel element) having non-uniform energy generation in axial direction is studied in detail. Accordingly, the two-dimensional heat conduction equation having a cosine variation of heat generation term in the axial direction, along with appropriate boundary conditions is solved using second order accurate finite difference schemes. The results are presented for a wide range of parameters like Aspect Ratio Ar, Biot number, Bi, Conductivity ratio, Cr, Thickness ratio, Th, and total heat generation parameter, Qt in the form of transverse temperature profiles and axial temperature profiles. Effect of Cr and Th due to presence of cladding is thoroughly studied. Different range of parameters for which the temperature of the fuel element is within the maximum temperature range is obtained. Critical values of these parameters for which the temperature increments above maximum temperature is also investigated. From the detailed discussion of the results it is concluded that the maximum temperature attained in the fuel element without cladding (bare fuel element) is very much higher than that attained in fuel element with cladding. Also it is observed that for fixed values of Ar, Cr and Th, there exists a critical value of Qt and Bi beyond and below which maximum temperature in the fuel element exceeds its allowable limit.

Numerical heat transfer analysis of Maisotsenko Humid Air Bottoming Cycle – A study towards the optimization of the air-water mixture at bottoming turbine inlet

Maisotsenko Humid Air Bottoming Cycle (MHABC) is a recently proposed waste heat recovery unit for gas turbine topping cycle. An air saturator is used for the waste heat recovery and air humidification purposes in the bottoming cycle. The humid air is used in bottoming cycle turbine. In this work, detailed heat transfer analysis of the MHABC is proposed along with the evaluation of degree of humidification and energy recovery. The air saturator is numerically simulated after the development of governing ordinary differential equations for the temperature and humidity distribution of the working air. The governing differential equations of air saturator are discretized using first order accurate finite difference scheme and simulated using MATLAB. The results for the air saturator are initially validated using inlet ambient pressure conditions at standard atmospheric conditions. The upper section of the bottoming cycle is modeled using analytical NTU method. The overall performance of the plant with MHABC is also compared with results reported in literature. Temperature and humidity levels for working air is presented. These results show that the proposed MHABC configuration gives maximum work output (58 MW) and efficiency (∼34%) at the air saturator exit relative humidity of 70%. Corresponding to the maximum system output conditions, the dew point effectiveness of the air saturator is noted at 0.8 with 32% relative humidity at the bottoming turbine inlet. While the air split ratio of 50% between lower air saturator and heat exchanger gives more than 8 MW of waste heat recovery.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak...

High order accurate finite difference schemes based on symmetry preservation

In this paper, we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve Lie symmetry groups of underlying partial differential equations (PDEs). In the proposed approach, invariant (or symmetry preserving) numerical schemes with a desired (or fixed) order of accuracy are constructed from some non-invariant (base) numerical schemes. Modified forms of PDEs are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of PDEs. These defect correction terms of modified PDEs that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames. This feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations. The proposed method is demonstrated via construction of invariant numerical schemes with fixed (and higher) order of accuracy for some common linear and nonlinear problems (including the linear advection–diffusion equation in 1D and 2D, inviscid Burgers' equation, and viscous Burgers' equation) and the performance of these invariant numerical schemes is further evaluated. Our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying PDEs.

Numerical Analysis on the Two-Dimensional Unsteady Magnetohydrodynamic Compressible Flow through a Porous Medium

In the present study, the unsteady magnetrohydrodynamic (MHD) flow of compressible fluid with variable thermal properties has been numerically investigated. The electrically conducting fluid flows through a porous media channel. The uniform magnetic field is applied perpendicular to the direction of the flow. The wall is assumed to be non-conducting and maintained at two different temperatures. The thermal conductivity and viscosity of the fluid change with temperature. Sixth - Order Accurate Compact Finite Difference scheme together with the Third-order Runge-Kutta method is used to solve a set of non-linear equations. The results of the calculation are expressed in the form of the velocity and temperature at different values of the magnetic field and porosity. The proposed mathematical model and numerical methods have been validated by comparing with the results of previously published studies that the compared results reveal the same trends. The difference is due to the compressibility and property variation effects. The results showed that the magnetic field and variable properties considerably influences the flows that is compressible thereby affecting the heat transfer as well as the wall shear stress.

Mixed Convection and Heat Transfer Studies in Non-Uniformly Heated Buoyancy Driven Cavity Flow

We analyse the mixed convection flow in a cavity flow which is driven by buoyancy generated due to a non-uniformly heated top wall which is moving uniformly. A fourth order accurate finite difference scheme is used in this study and our code is first validated against available data in the literature. The results are obtained for different sets of Reynolds number Re, Prandtl number Pr and Grashof number Gr which are in the ranges 100 - 3000, 0.0152 - 10 and 102 - 106 respectively. Here Gr is related to the Richardson number according to Ri=Gr/Re2. While increasing the Richardson number, the growth of upstream secondary eddy (USE) is observed together with a degradation of downstream secondary eddy (DSE). When mixed convection is dominant, the upstream secondary eddy and the downstream secondary eddy merge to form a large recirculation region. When the effect of Pr is studied in the forced convection regime, Ri<<1, the temperature in the central region of the cavity remains nearly a constant. However, in the mixed convection regime, the temperature in cavity undergoes non-monotonic changes. Finally, using the method of divided differences, it is shown that numerical accuracy of the derived numerical scheme used in this work is four.

A Family of Compact Finite Difference Formulations for Three-Space Dimensional Nonlinear Poisson’s Equations in Cartesian Coordinates

In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in three-space dimensions has been developed. The discretization formula is developed on a non-uniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of \(l^\infty \)- and \(l^2\)-error of the exact and approximate solution values.

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N?1 lnN)p),p=1,2, on the Shishkin mesh and O(N?p),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.

Stable and high order accurate difference methods for the elastic wave equation in discontinuous media

In this paper, we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media. We consider both planar and curved interfaces where media parameters are allowed to be discontinuous. The key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise. We discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule. Conditions at layer interfaces are imposed weakly using penalties. By deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability. We present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials. The numerical experiments verify high order accuracy and time stability.

Review of summation-by-parts schemes for initial–boundary-value problems

High-order finite difference methods are efficient, easy to program, scale well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback has been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years has removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area.

Asymptotic, numerical and approximate techniques for a free boundary problem arising in the diffusion of glassy polymers

This paper considers approximate solution methods for a one dimensional Stefan problem describing solvent diffusion in glassy polymers. Similar to the classic Stefan problem, the region initially has zero thickness and so must be analysed carefully before performing a numerical computation. A small-time analysis gives the correct starting solution which is then incorporated into the second order accurate Keller box finite difference scheme. We also consider a detailed analysis of small and large time expansions, as well as the large control parameter limit, and show that our generalised approach enables us to obtain higher order terms than given in previous studies. Finally, we apply the combined integral method (CIM) to this problem, which is a refinement of the popular heat balance integral method (HBIM), and compare both the CIM and asymptotic solutions to the numerical results.