Order Finite Difference Method Research Articles

A space marching method for sonic boom near field predictions

The conventional sonic boom propagation prediction method widely adopted in supersonic aircraft design involves a two-step procedure. In the first step, a compressible viscous or inviscid Computational Fluid Dynamics analysis is applied to the aircraft geometry at flight conditions to produce a flow field solution near the aircraft. Then in the second step, a one dimensional nonlinear Burgers’ equation model is used to propagate sonic boom signature traces from the near field solution at flight altitude to the ground along a ray path. For an accurate ground signature prediction the near field signature must be accurately modeled at a sufficient distance from the aircraft flight path in order to minimize errors in the one dimensional propagation model. This is a very challenging task for general purpose CFD tools in a design environment because the cost of maintaining highly accurate off-body solutions increases dramatically as the radial distance is enlarged in the computational domain. It is also particularly difficult to apply these tools for wave propagation because the algorithms are normally lower order and numerically dissipative and dispersive. In this work a space marching procedure based on an optimized higher order finite difference method is developed and applied in conjunction with a CFD solution concentrated in the close vicinity of the aircraft. This new approach is much more efficient, compared to previous methods, in providing highly accurate near field signatures for full carpet ground predictions. Results indicate that near field signatures retain more waveform shape information farther from the aircraft geometry while reducing the CFD cost significantly. The predicted ground signature is also shown to converge in shape as the radial distance of the near field signature grows, which is indicative of a more ideal initial condition being supplied to the one dimensional wave propagation to ground. This feature is very difficult to replicate in a tractable manner with common CFD approaches used in design.

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

A Higher Order Finite Difference Method for a Singularly Perturbed Boundary Value Problem with a Small Negative Shift

Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi

Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.

Analysis on mathematical model of convection system of micropolar fluid as darcy forchheimer flow undergoes heterogeneous and homogeneous chemical reaction

In this article, to study the fluid behavior on various specific conditions to improve the heat transfer system and here, analysis on mathematical model of dynamic fluid consist of micropolar fluid has allowed micro rotational effect with laminar flow of Darcy forchheimer model which allow inertia effect has incorporated with heterogeneous and homogenous chemical reaction undergoes heat exchanger system with boundary layer problem has modeled. The mathematical model of fluid mechanic governing equations are in the form of partial differential equation (PDE) and similarity transformation into numerical methods (PC4-FDM) of predictor and corrector technique undergoes discretized mesh point and convergence with fourth order finite difference method and shooting method is also equipped to get better solution. The additions of significant heterogeneous parameter depicts that increasing behavior in fluid concentration and homogeneous parameter depicts that decreasing in fluid concentration by allowed micro rotations leads to collision and on increasing the Eckert number related to viscous dissipation has exhibited that increased fluid velocity and decreased fluid temperature. Micro rotation parameter exhibits that similar increased fluid velocity and slight decreased in temperature of the fluid. Darcy forchheimer parameter which related to inertial effect has depicts that decreased in velocity with increased temperature of the fluid in convection system. Due to Industrialization, the study of convection heat transfer system has enormous scope which has necessity to improving the heating and cooling system of industrial mass machineries like powerplant, waste heat recovery unit, pharmaceutical industries etc.

A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

In this paper, a fourth order augmented matched interface and boundary (AMIB) method is proposed for solving a three-dimensional elliptic interface problem which involves a smooth material interface inside a cuboid domain. On the boundary of the cuboid domain, the fourth order AMIB method can handle different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations in fictitious value generation. Moreover, zero-padding solutions are introduced so that the fast Fourier transform (FFT) algorithm is still valid near the boundary. In dealing with the interior interface, a fourth order ray-casting matched interface and boundary (MIB) scheme is proposed, which enforces the jump conditions along the normal direction for calculating fictitious values. Comparing with the existing MIB scheme, the ray-casting scheme naturally bypasses the corner issue and becomes more robust in handling complex geometry. Based on fictitious values generated near interface and boundary, the fourth order central difference can be corrected at various irregular points including corner points, by introducing Cartesian derivative jumps as auxiliary variables. This gives rise to an enlarged linear system, which can be efficiently solved by the Schur complement procedure together with the FFT inversion of the discrete Laplacian. Extensive numerical experiments have been carried to test the proposed ray-casting AMIB method for numerical accuracy, efficiency, and robustness in corner treatment. The numerical results demonstrate that the ray-casting AMIB scheme not only maintains a fourth order of accuracy in treating various interfaces and boundaries for both solutions and solution gradients, but also attains an overall efficiency on the order of O(n3log⁡n) for a n×n×n uniform grid.

Modeling One Dimensional Two-Cell Model with Tumor Interaction Using Krylov Subspace Methods

A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.

On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

<p style='text-indent:20px;'>We consider an initial and Dirichlet boundary value problem for a logarithmic Schrödinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete <inline-formula><tex-math id="M1">\begin{document}$ L_t^{\infty}(L_x^2) $\end{document}</tex-math></inline-formula> norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrödinger equation is provided, without regularizing its nonlinear term.</p>

Higher order approximation in exponential form based on half-step grid-points for 2D quasilinear elliptic BVPs on a variant domain

This paper reports a new fourth order Finite Difference Method (FDM) in exponential form for two-dimensional quasilinear boundary value problem of elliptic type (BVPE) with variant solution domain. Further, this discretization is extended to solve the system of quasilinear BVPEs. Following are the main highlights of the proposed FDM:• An unequal mesh 9-point compact stencil is used to approximate the solution. Half-step points are used to evaluate the known variables of this problem. The convergence theory is studied for unequal mesh to validate the fourth order convergence of the suggested FDM.• It is applicable to BVPE irrespective of coordinate systems. Various benchmark problems, for example, Poisson equation in cylindrical coordinates, Burgers’ equation, Navier-Stokes (NS) equations in cylindrical and rectangular coordinates, are solved to depict their fourth order convergence.• Numerical results confirm the accuracy, trustworthiness and acceptability of the suggested numerical algorithm. These results endorse the superiority of the proposed FDM over the previously existing techniques of Mohanty and Kumar (2014), Mohanty and Setia (2014), Priyadarshini and Mohanty (2021).

A High Order Finite Difference Method for the Elastic Wave Equation in Bounded Domains with Nonconforming Interfaces

A High Order Finite Difference Method for the Elastic Wave Equation in Bounded Domains with Nonconforming Interfaces

An Extended Finite Difference Method for Singular Perturbation Problems on a Non-Uniform Mesh

An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.

Accelerating CFD simulation with high order finite difference method on curvilinear coordinates for modern GPU clusters

A high fidelity flow simulation for complex geometries for high Reynolds number (Re) flow is still very challenging, requiring a more powerful HPC system. However, the development of HPC with traditional CPU architecture suffers bottlenecks due to its high power consumption and technical difficulties. Heterogeneous architecture computation is raised to be a promising solution to the challenges of HPC development. GPU accelerating technology has been utilized in low order scheme CFD solvers on the structured grid and high order scheme solvers on unstructured meshes. The high-order finite difference methods on structured grids possess many advantages, e.g., high efficiency, robustness, and low storage. However, the strong dependence among points for a high-order finite difference scheme still limits its application on the GPU platform. In the present work, we propose a set of hardware-aware technology to optimize data transfer efficiency between CPU and GPU, as well as communication efficiency among GPUs. An in-house multi-block structured CFD solver with high order finite difference methods on curvilinear coordinates is ported onto the GPU platform and obtains satisfying performance with a speedup maximum of around 2000x over a single CPU core. This work provides an efficient solution to apply GPU computing in CFD simulation with specific high order finite difference methods on current GPU heterogeneous computers. The test shows that significant accelerating effects can be achieved for different GPUs.

Direct exchange calculation for unstructured micromagnetic meshes

In micromagnetic simulations it is necessary to compute the effective field associated to the exchange interaction, which corresponds to applying the Laplace operator to the magnetization field. The most widely used approach for computing the exchange field is to apply finite-difference schemes to homogeneous Cartesian meshes. In many situations it would be computationally advantageous to perform micromagnetic simulation on inhomogeneous meshes, but doing so presents a challenge with regard to the calculation of the exchange field. Here we present the construction of an exchange field calculation method for inhomogeneous meshes. A novel second order differential operator method is introduced and applied to a series of analytical functions on irregular meshes and compared to existing methods, finding an accuracy improvement of up to 81% relative to the second order finite difference method. The method is then used in combination with the MagTense framework and tested on the μmag standard micromagnetics problems 3 and 4. Here we find that especially for tetrahedral meshes, the proposed method results in much faster convergence as function of mesh size. We demonstrate that the method converges stably even for meshes where some mesh elements have a volume as small as 1/8 compared to the base resolution. Finally, we show that the method works equally well for meshes with spatially varying material parameters.

A Fft Accelerated Fourth Order Finite Difference Method for Solving Three-Dimensional Elliptic Interface Problems

A Fft Accelerated Fourth Order Finite Difference Method for Solving Three-Dimensional Elliptic Interface Problems

A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O(Nlog⁡N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.

Thermodynamically Consistent Algorithms for Models of Diblock Copolymer Solutions Interacting with Electric and Magnetic Fields

We derive thermodynamically consistent models for diblock copolymer solutions coupled with the electric and magnetic field, respectively. These models satisfy the second law of thermodynamics and are therefore thermodynamically consistent. We then design a set of 2nd order, linear, semi-discrete schemes for the models using the energy quadratization method and the supplementary variable method, respectively, which preserve energy dissipation rates of the models. The spatial discretization is carried out subsequently using 2nd order finite difference methods, leading to fully discrete, energy-dissipation-rate preserving algorithms that are thermodynamically consistent. Convergence rates are numerically confirmed through mesh refinement tests and several numerical examples are given to demonstrate the role of mobility in pattern formation, defect removing effect of both electric and magnetic fields as well as the hysteresis effect with respect to applied external fields in copolymer solutions.

Fourth order compact FD methods for convection diffusion equations with variable coefficients

Fourth order finite difference methods combined with an integrating factor strategy for steady convection and diffusion partial differential equations with variable coefficients in both 2D and 3D are proposed using uniform Cartesian grids. An integrating factor strategy is applied to transform the convection and diffusion PDE to a self-adjoint form. Then, a fourth order finite difference method is obtained through a second order scheme followed by the Richardson extrapolation. Another approach is a direct fourth order compact finite difference scheme. The developed integrating factor strategy provides an efficient way for dealing with large convection coefficients. Several numerical examples are presented to demonstrate the convergence order and compare the two fourth order methods.

Very high-order Cartesian-grid finite difference method on arbitrary geometries

An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin conditions. We introduce the Reconstruction Off-site Data (ROD) method, that transfers in polynomial functions the information located on the physical boundary to the computational domain. Three major advantages are: (1) a simple description of the physical boundary with Robin condition using a collection of points; (2) no analytical expression (implicit or explicit) is required, particularly the ghost cell centroids projection are not needed; (3) we split up into two independent machineries the boundary treatment and the resolution of the interior problem, coupled by the ghost cell values. Numerical evidences based on the simple 2D convection-diffusion operators are presented to prove the capability of the method to reach at least the 6th-order with arbitrary smooth domains.

Elastic Wave Propagation in Curvilinear Coordinates with Mesh Refinement Interfaces by a Fourth Order Finite Difference Method

We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the materia...

Sixth Order Finite Difference Method for Three Coupled Nonlinear Schrödinger Equations

In this work, we will derive a numerical method of sixth order in space and second order in time for solving 3-coupled nonlinear Schrödinger equations. The numerical method is unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we study the interaction dynamics of two solitons. It is found that both elastic and inelastic collisions can take place under suitable parametric conditions.