Point Stencil Research Articles

An Efficient Implicit Compact Streamfunction Velocity Formulation of Two Dimensional Flows

In this paper, we develop a class of implicit, unconditionally stable compact schemes for solving coupled fourth order and second order semilinear streamfunction and temperature equations respectively representing unsteady Navier---Stokes (N---S) equations for incompressible viscous fluid flows. The governing equations are presented in a generic form from which specific equations are obtained for several two dimensional (2D) incompressible viscous fluid flow problems. The streamfunction and the temperature equations are all solved iteratively in a coupled system of equations for the four field variables consisting of streamfunction, two velocities and temperature. The discretized form of biharmonic equation in streamfunction consists on 5-point stencil of streamfunction and is found to be second-order accurate both for spatially and temporally. In addition, we have considered two strategies for the discretization of the energy equation, which are second order accurate 5-point and fourth order accurate 9-point compact scheme. The formulation is made efficient by decreasing the computational complexity and is used to solve several 2D time dependent well studied benchmark problems. It is seen to efficiently capture both time wise and steady-state solutions of the N---S equations with Dirichlet as well as Neumann boundary conditions. The results obtained using our proposed scheme are in excellent agreement with the results available in the literature and they clearly establish the efficiency and the accuracy of the proposed scheme.

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

AbstractImage registration is an ill-posed problem that has been studied widely in recent years. The so-called curvature-based image registration method is one of the most effective and well-known approaches, as it produces smooth solutions and allows an automatic rigid alignment. An important outstanding issue is the accurate and efficient numerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonic equations, addressed in this article. We propose a fourth-order compact (FOC) finite difference scheme using a splitting operator on a 9-point stencil, and discuss how the resulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operator involved by a local Fourier analysis (LFA), we show that our FOC finite difference method is amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure. A high potential point-wise smoother using an outer-inner iteration method is shown to be effective by the LFA and numerical experiments. Real medical images are used to compare the accuracy and efficiency of our approach and the standard second-order central (SSOC) finite difference scheme in the same NMG framework. As expected for a higher-order finite difference scheme, the images generated by our FOC finite difference scheme prove significantly more accurate than those computed using the SSOC finite difference scheme. Our numerical results are consistent with the LFA analysis, and also demonstrate that the NMG method converges within a few steps.

A high-order time formulation of the RBC schemes for unsteady compressible Euler equations

Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5×5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock.

ChipPCR: an R package to pre-process raw data of amplification curves.

Both the quantitative real-time polymerase chain reaction (qPCR) and quantitative isothermal amplification (qIA) are standard methods for nucleic acid quantification. Numerous real-time read-out technologies have been developed. Despite the continuous interest in amplification-based techniques, there are only few tools for pre-processing of amplification data. However, a transparent tool for precise control of raw data is indispensable in several scenarios, for example, during the development of new instruments. chipPCR is an R: package for the pre-processing and quality analysis of raw data of amplification curves. The package takes advantage of R: 's S4 object model and offers an extensible environment. chipPCR contains tools for raw data exploration: normalization, baselining, imputation of missing values, a powerful wrapper for amplification curve smoothing and a function to detect the start and end of an amplification curve. The capabilities of the software are enhanced by the implementation of algorithms unavailable in R: , such as a 5-point stencil for derivative interpolation. Simulation tools, statistical tests, plots for data quality management, amplification efficiency/quantification cycle calculation, and datasets from qPCR and qIA experiments are part of the package. Core functionalities are integrated in GUIs (web-based and standalone shiny applications), thus streamlining analysis and report generation. http://cran.r-project.org/web/packages/chipPCR. Source code: https://github.com/michbur/chipPCR. stefan.roediger@b-tu.de Supplementary data are available at Bioinformatics online.

A comparison of CPU and GPU implementations for solving the Convection Diffusion equation using the local Modified SOR method

In this paper we study a parallel form of the SOR method for the numerical solution of the Convection Diffusion equation suitable for GPUs using CUDA. To exploit the parallelism offered by GPUs we consider the fine grain parallelism model. This is achieved by considering the local relaxation version of SOR. More specifically, we use SOR with red-black ordering using two sets of parameters @w1ij and @w2ij for the 5 point stencil. The parameter @w1ij is associated with each red (i+j even) grid point (i,j), whereas the parameter @w2ij is associated with each black (i+j odd) grid point (i,j). The use of a parameter for each grid point avoids the global communication required in the adaptive determination of the best value of @w and also increases the convergence rate of the SOR method (Varga, 1962) [38] and (Young, 1971) [41]. We present our strategy and the results of our effort to exploit the computational capabilities of GPUs under the CUDA environment. Additionally, two parallel CPU programs utilizing manual SSE2 (Streaming SIMD Extensions 2) and AVX (Advanced Vector Extensions) vectorization were developed as performance references. The optimizations applied on the GPU version were also considered for the CPU version. Significant performance improvement was achieved with all three developed GPU kernels differentiated by the degree of recomputations thus affecting the flops per element access ratio.

Solving global problem by considering multitude of local problems: Application to fluid flow in anisotropic porous media using the multipoint flux approximation

In this work we apply the experimenting pressure field approach to the numerical solution of the single phase flow problem in anisotropic porous media using the multipoint flux approximation. We apply this method to the problem of flow in saturated anisotropic porous media. In anisotropic media the component flux representation requires, generally multiple pressure values in neighboring cells (e.g., six pressure values of the neighboring cells is required in two-dimensional rectangular meshes). This apparently results in the need for a nine points stencil for the discretized pressure equation (27 points stencil in three-dimensional rectangular mesh). The coefficients associated with the discretized pressure equation are complex and require longer expressions which make their implementation prone to errors. In the experimenting pressure field technique, the matrix of coefficients is generated automatically within the solver. A set of predefined pressure fields is operated on the domain through which the velocity field is obtained. Apparently such velocity fields do not satisfy the mass conservation equations entailed by the source/sink term and boundary conditions from which the residual is calculated. In this method the experimenting pressure fields are designed such that the residual reduces to the coefficients of the pressure equation matrix.

Calculation of the local rupture speed of dynamically propagating earthquakes

The velocity at which a propagating earthquake advances on the fault surface is of pivotal importance in the contest of the source dynamics and in the modeling of the ground motions generation. In this paper the problem of the determination of the rupture speed (<em>v_<sub>r</sub></em>) is considered. The comparison of different numerical schemes to compute <em>v<sub>r</sub></em> from the rupture time (<em>t_<sub>r</sub></em>) shows that, in general, central finite differences schemes are more accurate than forward or backward schemes, regardless the order of accuracy. Overall, the most efficient and accurate algorithm is the five–points stencil method at the second–order of accuracy. It is also shown how the determination of <em>t_<sub>r</sub></em> can affect <em>v<sub>_r </sub></em>; numerical results indicate that if the fault slip velocity threshold (<em>v_<sub>l</sub></em>) used to define <em>t_<sub>r</sub></em> is too high (<em>v<sub>_l</sub></em> ≥ 0.1 m/s) the details of the rupture are missed, for instance the rupture tip bifurcation occurring for 2–D supershear rupture. On the other hand, for <em>v_<sub>l</sub></em> ≤ 0.01 m/s the results appear to be stable and independent on the choice of <em>v_<sub>l </sub></em>. Finally, it is demonstrated that in the special case of the linear slip–weakening friction law the definitions of <em>t_<sub>r</sub></em> from the threshold criterion on the fault slip velocity and from the achievement of the maximum yield stress are practically equivalent.

High order accurate and low dissipation method for unsteady compressible viscous flow computation on helicopter rotor in forward flight

An improved fifth-order weighted essentially non-oscillatory (WENO-Z) scheme combined with the moving overset grid technique has been developed to compute unsteady compressible viscous flows on the helicopter rotor in forward flight. In order to enforce periodic rotation and pitching of the rotor and relative motion between rotor blades, the moving overset grid technique is extended, where a special judgement standard is presented near the odd surface of the blade grid during search donor cells by using the Inverse Map method. The WENO-Z scheme is adopted for reconstructing left and right state values with the Roe Riemann solver updating the inviscid fluxes and compared with the monotone upwind scheme for scalar conservation laws (MUSCL) and the classical WENO scheme. Since the WENO schemes require a six point stencil to build the fifth-order flux, the method of three layers of fringes for hole boundaries and artificial external boundaries is proposed to carry out flow information exchange between chimera grids. The time advance on the unsteady solution is performed by the full implicit dual time stepping method with Newton type LU-SGS subiteration, where the solutions of pseudo steady computation are as the initial fields of the unsteady flow computation. Numerical results on non-variable pitch rotor and periodic variable pitch rotor in forward flight reveal that the approach can effectively capture vortex wake with low dissipation and reach periodic solutions very soon.

A compact 9 point stencil based on integrated RBFs for the convection–diffusion equation

In this paper, a high-order compact stencil for solving the convection–diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over 3×3 rectangular stencils. Salient features here are that (i) integration is employed to construct local RBF approximations; and (ii) through the constants of integration, values of the convection–diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (i) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ii) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (iii) high levels of accuracy are achieved using relatively coarse grids.

A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations

AbstractIn this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values ofu,(∂2u/∂n2) and (∂4u/∂n4) are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

A new family of (5,5)CC-4OC schemes applicable for unsteady Navier–Stokes equations

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection–diffusion equation with variable convection coefficient is proposed. The schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady Navier–Stokes system. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Padé scheme for spatial derivative. These schemes which are based on a five point stencil with constant coefficients, named as “(5,5) Constant Coefficient 4th Order Compact” [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary conditions. Subsequently the proposed schemes are applied to problems governed by the incompressible Navier–Stokes equations. The results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed schemes.

A virtual node algorithm for Hodge decompositions of inviscid flow problems with irregular domains

We present an efficient discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions using a virtual node framework. The method is designed for use in the exact projection discretization of incompressible flow. We leverage the Poisson framework initially developed in [1] and [2]. This approach uses a signed distance function to represent the irregular domain embedded in a Cartesian grid and uses a variational approach to create a symmetric positive definite linear system. We present a novel modification to the previous approach that yields a 5-point stencil (7-point in 3D) across the entire computational domain. The original algorithm required a 9-point stencil (27-point in 3D) near the embedded irregular boundary. We show that this new condensed stencil enables a decomposition of the form A = G T M 1 G, where M is a diagonal weighting matrix and G and D = G T are diagonal scalings of the standard central-difference gradient and divergence operators. We use this factored form as the basis of our discrete Hodge decomposition and show that this can be readily used for exact projection in incompressible flow. Numerical experiments suggest our method is second-order in L ¥ for pressures and first-order in L ¥ , second-order in L 1 for velocities.

A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

In this paper, a compact 5-point stencil for the discretisation of second-order partial differential equations (PDEs) in two space dimensions is proposed. We employ integrated radial basis functions in one dimension (1D-IRBFs) to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil. Certain nodal values of the second-order derivatives are incorporated into the approximations with the help of the integration constants. In the case of elliptic PDEs, one algebraic equation is formed at each interior node, and the obtained final system, of which each row has 5 non-zero entries, is solved iteratively using a Picard scheme. In the case of parabolic PDEs discretised with a Crank–Nicolson procedure, a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach. Linear and non-linear test problems, including lid-driven cavity flow and natural convection between the outer square and the inner cylinder, are considered to verify the proposed stencil.

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr=0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 103 to 105 and inclination angles γ (0°⩽γ<360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.

A transport-physics-accurate convection discretization based on first order upwind and the need for a new error assessment strategy

Several numerical schemes have been proposed in the literature to deal with convection phenomena, but all of them present some spurious effects. In this paper we show that, once a more physically-oriented error assessment tool is used in substitution for the classical wave number analysis, it is possible to develop an improved family of numerical schemes. Results are presented for a particular implementation based on first order upwind, in test cases involving linear and non-linear convective equation. This new scheme is able to simulate linear convection equation under uniform field with machine accuracy precision, with zero diffusion and dispersion. Moreover, the results for the non-linear convection are also significantly more accurate over the whole spectrum when compared to a traditional scheme based on a fifteen points stencil.

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.

A Correction Function Method for Poisson problems with interface jump conditions

In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard “black box” solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the “standard” approaches used to compute the GFM correction terms. In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.

Exact evaluation of limits and tangents for interpolatory subdivision surfaces at rational points

This paper presents a new method for exact evaluation of a limit surface generated by stationary interpolatory subdivision schemes and its associated tangent vectors at arbitrary rational points. The algorithm is designed on the basis of the parametric m -ary expansion and construction of the associated matrix sequence. The evaluation stencil of the control points on the initial mesh is obtained, through computation, by multiplying the finite matrices in a sequence corresponding to the expansion sequence and eigendecomposition of the contractive matrix related to the period of rational numbers. The method proposed in this paper works for other non-polynomial subdivision schemes as well.

Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

AbstractThis paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

Three‐dimensional finite difference saturated‐unsaturated flow modeling with nonorthogonal grids using a coordinate transformation method

Study of the saturated‐unsaturated flow in porous media is of interest in many branches of science and engineering. Among the various numerical simulation methods available, the finite difference method is advantageous because it offers simplicity of discretization. This method has been widely used for simulating saturated‐unsaturated flows. However, the simulation of geometrically complex flow domains requires the use of high‐resolution grids in conventional finite difference models because conventional finite difference discretization assumes an orthogonal coordinate system. This makes a finite difference model computationally less efficient than other numerical models that can treat nonorthogonal grids, such as the finite element model and finite volume model. To overcome this disadvantage, we use a coordinate transformation method and develop a multidimensional finite difference model for simulating saturated‐unsaturated flows; this model can treat nonorthogonal grids. The cross‐derivative terms derived by the coordinate transformation method are evaluated explicitly for ease of coding. Therefore, a 7 point stencil is used for implicit terms in the iterative calculation, as in the case of conventional finite difference models with an orthogonal grid. We assess the performance of the proposed model by carrying out test simulations. We then compare the simulation results with dense grid solutions in order to evaluate the numerical accuracy of the proposed model. To examine the performance of the proposed model, we draw a comparison between the simulation results obtained using the proposed model and the results obtained by using (1) a model in which all terms are considered fully implicitly, (2) a finite element model, and (3) a conventional finite difference model with a high‐resolution orthogonal grid.