Point Stencil Research Articles

A high accuracy compact semi-constant mesh off-step discretization in exponential form for the solution of non-linear elliptic boundary value problems

This paper describes a novel implicit method of order 2 in y- and 3 in x-direction in exponential form, by exploiting off-step discretization to solve numerically non-linear elliptic partial differential equations in a rectangular region. We use variable mesh in x-direction and constant mesh in y-direction in order to solve convection–diffusion equation for large values of the coefficient of convection term. This method uses 9-point compact stencil. Detailed derivation and convergence procedure of the proposed method have been discussed. The method has been generalized to solve non-linear elliptic equations in vector form. The method is validated on several benchmark problems showing that formulation produces satisfactory results.

Discretization of Laplacian Operator in Polar Coordinates System on 9-Point Stencil with Mixed PDE’s Derivative Approximation Using Finite Difference Method

Discretization of Laplacian Operator in Polar Coordinates System on 9-Point Stencil with Mixed PDE’s Derivative Approximation Using Finite Difference Method

Common HPC Approaches in Python Evaluated for a Scientific Computing Test Case

A number of the most common high-performance computing approaches available in the Python programming environment of the LNCC Santos Dumont supercomputer, are compared using a specific test case. Python includes specific libraries, development tools, implementations, documentation and optimizing/parallelizing resources. It provides a straightforward way to program in a high level of abstraction, but parallelization resources to exploit multiple cores, processors or accelerators like GPUs are diverse and may be not easily selectable by the programmer. This work makes a comparison of common approaches in Python to boost computing performance. The test case is a well-known 2D heat transmission problem modeled by the Poisson partial-differential equation, which is solved by a finite difference method that requires the calculation of a 5-point stencil over the domain grid. Their serial and parallel implementations in Fortran 90 were taken as references in order to compare their performance to some serial and parallel Python implementations of the same algorithm. Besides performance results, a discussion about the trade-off between easiness of programming versus processing performance is included. This work is a primer for the use of HPC resources in Python.

A new ninth‐order central Hermite weighted essentially nonoscillatory scheme for hyperbolic conservation laws

AbstractIn this article, we propose a new ninth‐order central Hermite weighted essentially nonoscillatory (HWENO) scheme, for solving hyperbolic conservation laws. The new scheme consists of the following: ninth‐order reconstruction using only five points stencil; to calculate the linear weights we used the central WENO (CWENO) technique and for nonlinear weights we used a new weighting technique. The numerical solution is advanced in time by using the ninth‐order linear strong‐stability‐preserving Runge–Kutta (ℓSSPRK) scheme and for computing the numerical flux, we used the central‐upwind flux which is efficient, simple and can be used for nonconex fluxes problems. The resulting scheme is ninth order in both smooth regions and at critical points with very small numerical dissipation near discontinuities, this is due to using new smoothness indicators. Several numerical examples are presented for one‐ and two‐dimensional problems to confirm that the new scheme is superior to the other high‐order WENO schemes.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—part 2: numerical simulations and comparison with FEM

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( $$0.1 h - 10^{-9}h$$ where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

Energy and performance improvements in stencil computations on multi-node HPC systems with different network and communication topologies

Energy and performance improvements in stencil computations are relevant for both application developers and data center administrators. They appear as the fundamental scheme in many large-scale scientific simulations and workloads. Many research efforts have focused on some estimation techniques of the energy usage of HPC systems based on specific characteristics of parallel applications. In case of stencils, we have previously concentrated on detailed estimations of energy consumption and the energy-aware distribution of stencil computations on heterogeneous processors. However, we have restricted our comprehensive studies to a single heterogeneous computing node only. In this paper, we show how scheduling and optimization techniques can be applied for energy and performance improvements of stencil computations on multi-node HPC systems using different network topologies. We formulate a scheduling model together with a new Tabu Search algorithm, called Task Movement (TM), taking into account the communication hierarchies, to minimize the overall energy usage and the execution time of stencil computations. Experimental studies show that this algorithm solves the considered problem more efficiently comparing to other, simpler heuristics. We present computational experiments for a reference 7 point stencil computation pattern on three commonly used low-diameter network topologies: Fat-tree, Dragonfly, and Torus. According to our studies, the most promising multi-node HPC architecture for stencil computations is based on the Torus network concept. Finally, we argue that the proposed scheduling model and TM algorithm can be easily adopted within existing high-level parallel execution environments for stencils automatic performance tuning.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

A high-order numerical approach with Cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials

Recently we have developed a new numerical approach for PDEs with constant coefficients on irregular domains and Cartesian meshes. In this paper we extend it to a much more general case of PDEs with variable coefficients that have a lot of applications; e.g., the modeling of functionally graded materials, the inhomogeneous materials obtained by 3-D printing and many others. Here, we consider the 2-D wave and heat equations for isotropic and anisotropic inhomogeneous materials. The idea of the extension to the case of PDEs with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size. This leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach; however, this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique. Similar to our previous technique, the new 2-D approach with compact 9-point stencils uses trivial Cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. At similar 9-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the third order) finite elements with much wider stencils. The wave and heat equations are uniformly treated with the new approach.

A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshes

A new numerical approach for the time independent Helmholtz equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. At similar 9-point stencils, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions than that for the linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

Solving elliptic interface problems with jump conditions on Cartesian grids

We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the L∞-norm on Cartesian grids. The condition number is bounded, regardless of the ratio of the diffusion constant and scales like that of the standard 5-point stencil approximation on a rectangular grid with no interface. Numerical examples are given in two and three spatial dimensions.

Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

A new approach for the increase in the order of accuracy of high-order numerical techniques used for 2-D time-dependent elasticity (structural dynamics and wave propagation) has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding semi-discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of the linear finite elements with 9-point stencils is optimal and cannot be improved. However, we have calculated new 9-point stencils with the second order of accuracy that yield more accurate results than those obtained by the linear finite elements. We have also developed new 25-point stencils (similar to those for the quadratic finite and isogeometric elements) with the optimal sixth order of accuracy for the non-diagonal mass matrix and with the optimal fourth order of accuracy for the diagonal mass matrix. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for the high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by the high-order (up to the eighth order) finite elements with the non-diagonal mass matrix. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equations

A new approach for the increase in the order of accuracy of high-order numerical techniques used for time-independent elasticity is suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of 2-D linear finite elements with 9-point stencils is optimal and cannot be improved. However, the order of accuracy of 25-point stencils (similar to those for quadratic finite and isogeometric elements) can be significantly improved. We have developed new 25-point stencils for 2-D elastic problems with optimal 10th order of accuracy. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by high-order (up to the tenth order) finite elements. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Compact Approximate Taylor Methods for Systems of Conservation Laws

A new family of high order methods for systems of conservation laws is introduced: the Compact Approximate Taylor (CAT) methods. As in the approximate Taylor methods proposed by Zorio et al. (J Sci Comput 71(1):246–273, 2017) the Cauchy–Kovalevskaya procedure is circumvented by using Taylor approximations in time that are computed in a recursive way. The difference is that here this strategy is applied locally to compute the numerical fluxes what leads to methods that have $$(2 p +1)$$ -point stencil and order of accuracy 2p, where p is an arbitrary integer. Moreover we prove that they reduce to the high-order Lax–Wendroff methods for linear problems and hence they are linearly $$L^2$$ -stable under a $$CFL-1$$ condition. In order to prevent the spurious oscillations that appear close to discontinuities two shock-capturing techniques have been considered: a flux-limiter technique and WENO reconstruction for the first time derivative (WENO-CAT methods). We follow [25] in the second approach. A number of test cases are considered to compare these methods with other WENO-based schemes: the linear transport equation, Burgers equation, the 1D compressible Euler equations, and the ideal Magnetohydrodynamics equations are considered. Although CAT methods present an extra computational cost due to the local character, this extra cost is compensated by the fact that they still give good solutions with CFL values close to 1.

A higher-order blended compact difference (BCD) method for solving the general 2D linear second-order partial differential equation

A higher-order blended compact difference (BCD) scheme is proposed to solve the general two-dimensional (2D) linear second-order partial differential equation. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. Sixth-order accuracy approximations for the first- and second-order derivatives are employed, and the original equation is also discretized based on a 9-point stencil, which is different from the work of Lee et al. (J. Comput. Appl. Math. 264:23–37, 2014). A truncation error analysis is performed to show that the scheme is of sixth-order accuracy for the interior grid points. Simultaneously, sixth-order accuracy schemes are proposed to compute the grid points on the boundaries for the first- and second-order derivatives. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the present method.

Implementation of a 9-point stencil in SOLPS-ITER and implications for Alcator C-Mod divertor plasma simulations

The SOLPS-ITER code suite is used worldwide for plasma edge modeling, the interpretation of experiments, as well as for the design of the ITER divertor. The numerical scheme of the plasma solver of the code, B2.5, is based on the assumption of perfectly orthogonal grids in the poloidal plane, aligned with the magnetic field, while in practice grids are often strongly distorted to match divertor target shapes. Neglecting these grid distortion leads to qualitatively and quantitatively incorrect results for fluid neutral simulations, and may affect results in cold (detached) divertors even when using kinetic neutral simulations. In this contribution, we present the first results of a newly implemented 9-point stencil in B2.5 to properly handle misaligned grids. The new scheme is then applied to fluid neutral simulations of a well-diagnosed and previously modeled Alcator C-Mod discharge. Results are compared with the original 5-point scheme neglecting grid distortion effects, as well as with simulations including a full kinetic neutral model. We conclude that the 9-point stencil is essential to correctly model the transport of fluid neutrals on distorted grids, and to capture the effects of divertor closure on the fluid neutral behavior.

Numerical solution of 2D Navier–Stokes equation discretized via boundary elements method and finite difference approximation

In this paper boundary elements method (BEM) is equipped with finite difference approximation (FDA) to solve two-dimensional Navier–Stokes (N–S) equation. The N–S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, forward Euler, Runge–Kutta and Newton’s methods to find a fast ODE solver. Numerical investigations presented in this article show Newton’s method is faster than the others and it is able to solve the N–S equation for Reynolds number up to 40,000 when grid points are at most 141 × 141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM’s two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly agree with that accurate ones available in the literature.

A Finite Element Galerkin High Order Filter for the Spherical Limited Area Model

Two dimensional finite element method with quadrilateral basis functions was applied to the spherical high order filter on the spherical surface limited area domain. The basis function consists of four shape functions which are defined on separate four grid boxes sharing the same gridpoint. With the basis functions, the first order derivative was expressed as an algebraic equation associated with nine point stencil. As the theory depicts, the convergence rate of the error for the spherical Laplacian operator was found to be fourth order, while it was the second order for the spherical Laplacian operator. The accuracy of the new high order filter was shown to be almost the same as those of Fourier finite element high order filter. The two-dimension finite element high order filter was incorporated in the weather research and forecasting (WRF) model as a hyper viscosity. The effect of the high order filter was compared with the built-in viscosity scheme of the WRF model. It was revealed that the high order filter performed better than the built in viscosity scheme did in providing a sharper cutoff of small scale disturbances without affecting the large scale field. Simulation of the tropical cyclone track and intensity with the high order filter showed a forecast performance comparable to the built in viscosity scheme. However, the predicted amount and spatial distribution of the rainfall for the simulation with the high order filter was closer to the observed values than the case of built in viscosity scheme.

A-posteriori error estimation for the finite point method with applications to compressible flow

An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented.

An efficient high-order compact scheme for the unsteady compressible Euler and Navier–Stokes equations

Residual-Based Compact (RBC) schemes approximate the 3-D compressible Euler equations with a 5th- or 7th-order accuracy on a 5×5×5-point stencil and capture shocks pretty well without correction. For unsteady flows however, they require a costly algebra to extract the time-derivative occurring at several places in the scheme. A new high-order time formulation has been recently proposed [13] for simplifying the RBC schemes and increasing their temporal accuracy. The present paper goes much further in this direction and deeply reconsiders the method. An avatar of the RBC schemes is presented that greatly reduces the computing time and the memory requirements while keeping the same type of successful numerical dissipation. Two and three-dimensional linear stability are analyzed and the method is extended to the 3-D compressible Navier–Stokes equations. The new compact scheme is validated for several unsteady problems in two and three dimension. In particular, an accurate DNS at moderate cost is presented for the evolution of the Taylor–Green Vortex at Reynolds 1600 and Prandtl 0.71. The effects of the mesh size and of the accuracy order in the approximation of Euler and viscous terms are discussed.

An unconditionally monotone numerical scheme for the two-factor uncertain volatility model

Under the assumption that two asset prices follow an uncertain volatility model, the maximal and 2 minimal solution values of an option contract are given by a two dimensional Hamilton-Jacobi-Bellman 3 (HJB) PDE. A fully implicit, unconditionally monotone nite dierence numerical scheme is developed 4 in this paper. Consequently, there are no time step restrictions due to stability considerations. The 5 discretized algebraic equations are solved using policy iteration. Our discretization method results in a 6 local objective function which is a discontinuous function of the control. Hence some care must be taken 7 when applying policy iteration. The main diculty in designing a discretization scheme is development 8 of a monotonicity preserving approximation of the cross derivative term in the PDE. We derive a hybrid 9 numerical scheme which combines use of a xed point stencil and a wide stencil based on a local coordinate 10 rotation. The algorithm uses the xed point stencil as much as possible to take advantage of its accuracy 11 and computational eciency. The analysis shows that our numerical scheme is l1 stable, consistent in 12 the viscosity sense, and monotone. Thus, our numerical scheme guarantees convergence to the viscosity 13 solution. 14