Wide Stencil Research Articles

Robust and accurate Roe-type Riemann solver with compact stencil: Rotated-RoeM scheme

In this study, we aim to develop a robust and accurate Riemann solver to resolve strong shock waves with compact stencil information by applying the rotated hybrid concept to the RoeM scheme. The original RoeM scheme offers good robustness from the perspective of avoiding shock instability, also called the carbuncle phenomenon, in hypersonic flows by introducing Mach-number-based functions. These functions rely on multi-dimensional shock discontinuity sensing terms to capture grid-aligned shock waves, thus necessitating sharing of values of cells adjacent to each vertex. Consequently, constructing a data structure that can handle such a wide stencil is necessary, particularly for implementations on unstructured grids. Moreover, communication of this stencil information on parallel computation can increase computational costs.To overcome these complexities, the rotated hybrid concept is applied to the RoeM scheme, to replace the multi-dimensional terms while still resolving the grid-aligned shock instability. In addition, a Mach-number-based weighting function is introduced to distinguish between the flow around the boundary layer and the onset of shock instability. The damping characteristic of the proposed scheme is compared to that of the original RoeM scheme by conducting a linear perturbation analysis. The findings reveal that the proposed scheme effectively controls spurious oscillations across strong shock waves. Numerous test cases are presented to demonstrate the robustness and accuracy of the proposed approach. Notably, the proposed scheme successfully resolves thermal boundary layers and accurately predicts the heat transfer rate around stagnation points in hypersonic flows.

11-th order of accuracy for numerical solution of 3-D Poisson equation with irregular interfaces on unfitted Cartesian meshes

For the first time the optimal local truncation error method (OLTEM) with 125-point stencils and unfitted Cartesian meshes has been developed in the general 3-D case for the Poisson equation for heterogeneous materials with smooth irregular interfaces. The 125-point stencils equations that are similar to those for quadratic finite elements are used for OLTEM. The interface conditions for OLTEM are imposed as constraints at a small number of interface points and do not require the introduction of additional unknowns, i.e., the sparse structure of global discrete equations of OLTEM is the same for homogeneous and heterogeneous materials. The stencils coefficients of OLTEM are calculated by the minimization of the local truncation error of the stencil equations. These derivations include the use of the Poisson equation for the relationship between the different spatial derivatives. Such a procedure provides the maximum possible accuracy of the discrete equations of OLTEM. In contrast to known numerical techniques with quadratic elements and third order of accuracy on conforming and unfitted meshes, OLTEM with the 125-point stencils provides 11-th order of accuracy, i.e., an extremely large increase in accuracy by 8 orders for similar stencils. The numerical results show that OLTEM yields much more accurate results than high-order finite elements with much wider stencils. The increased numerical accuracy of OLTEM leads to an extremely large increase in computational efficiency.Also, a new post-processing procedure with the 125-point stencil has been developed for the calculation of the spatial derivatives of the primary function. The post-processing procedure includes the minimization of the local truncation error and the use of the Poisson equation. It is demonstrated that the use of the partial differential equation (PDE) for the 125-point stencils improves the accuracy of the spatial derivatives by 6 orders compared to post-processing without the use of PDE as in existing numerical techniques. At an accuracy of 0.1% for the spatial derivatives, OLTEM reduces the number of degrees of freedom by 900−4⋅106 times compared to quadratic finite elements. The developed post-processing procedure can be easily extended to unstructured meshes and can be independently used with existing post-processing techniques (e.g., with finite elements).

Sliding or Stumbling on the Staircase: Numerics of Ocean Circulation Along Piecewise‐Constant Coastlines

AbstractCoastlines in most ocean general circulation models are piecewise constant. Accurate representation of boundary currents along staircase‐like coastlines is a long‐standing issue in ocean modeling. Pioneering work by Adcroft and Marshall (1998, https://doi.org/10.3402/tellusa.v50i1.14514) revealed that artificial indentation of model coastlines, obtained by rotating the numerical mesh within an idealized square basin, generates a spurious form drag that slows down the circulation. Here, we revisit this problem and show how this spurious drag may be eliminated. First, we find that physical convergence to spatial resolution (i.e., the main characteristics of the flow are insensitive to the increase of the mesh resolution) allows simulations to become independent of the mesh orientation. An advection scheme with a wider stencil also reduces sensitivity to mesh orientation from coarser resolution. Second, we show that indented coastlines behave as straight and slippery shores when a true mirror boundary condition on the flow is imposed. This finding applies to both symmetric and rotational‐divergence formulations of the stress tensor, and to both flux and vector‐invariant forms of the equations. Finally, we demonstrate that the detachment of a vortex flowing past an outgoing corner of the coastline is missed with a free‐slip (zero vorticity) condition at the corner. These results provide guidance for a better numerical treatment of coastlines (and isobaths) in ocean general circulation models.

Optimal local truncation error method for solution of 2-D elastodynamics problems with irregular interfaces and unfitted Cartesian meshes as well as for post-processing

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation.

A novel high-order solver for simulation of incompressible flows using the flux reconstruction method and lattice Boltzmann flux solver

In this paper, a novel high-order solver using the flux reconstruction (FR) method and the lattice Boltzmann flux solver (LBFS) is proposed for accurately and efficiently simulating nearly incompressible flows. The existing LBFSs are all based on the finite volume (FV) scheme. The present work extends the application of the LBFS for the first time by combining the high-order FR scheme. Compared with the traditional high-order incompressible Navier-stokes (N-S) solvers, no particular techniques are needed for the current FR-LBFS to overcome the difficulty of the pressure-velocity coupling since the present method is a weakly compressible model. Moreover, unlike the traditional N-S solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly in FR-LBFS. In the present method, the common inviscid and viscous fluxes at the cell interface of the FR scheme are evaluated simultaneously by the local reconstruction of lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Thus the discretization of the second-order partial derivative term is avoided. Compared with the high-order off-lattice Boltzmann methods (OLBM), the FR-LBFS is more stable, efficient and low-storage. No special techniques are needed to remove the stiffness of the governing equations which existing in the discrete velocity Boltzmann equation (DVBE). Compared with the high-order FV-LBFS, the FR-LBFS is compact for parallel computing by avoiding wide stencils on meshes. In addition, the present scheme has lower dissipation and can be more flexible to increase accuracy order. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) steady plane Poiseuille flow, (c) lid-driven cavity flow, (d) laminar boundary layer and (e) flow past a square cylinder.

A combined flux reconstruction and lattice Boltzmann flux solver for inviscid compressible flow simulation

AbstractA numerical method, which combines the high order flux reconstruction (FR) and the lattice Boltzmann flux solver (LBFS), has been developed to solve inviscid compressible flows in this article. Extensive research has shown that LBFS has absorbed the advantages of both finite volume method (FVM) and lattice Boltzmann method (LBM). On one hand, compared with FVM, FR can achieve arbitrary high order accuracy and is compact for parallel computing by avoiding wide stencils on meshes, which shows FR is a better choice than FVM. On the other hand, different from the discrete velocity Boltzmann equation based methods, the present method, that is, FR‐LBFS, solves the Euler equations essentially, and then the calculation efficiency can be improved. In FR‐LBFS, the common inviscid flux at the cell interface is evaluated by the local reconstruction of the lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Unlike the traditional flux solution at the interface where the nonlinear convective terms have to be considered in the computation, the feature of linear convection term in LBE can make it easier to calculate flux in the current method. Furthermore, by considering real physical effects in LBM, the positive density and pressure can be achieved in the case of strong shocks and discontinuities. Some typical inviscid problems, including the advection of density perturbation, sod shock tube, Riemann problem for two‐dimensional gas dynamics, supersonic flow in a convergent nozzle with a ramp, subsonic flow around a cylinder, and transonic flow over a NACA0012 airfoil, are simulated to demonstrate the accuracies and robustness of the proposed FR‐LBFS.

High-order compact difference schemes on wide computational stencils with a spectral-like accuracy

In this paper, we propose a generalized framework for the derivation of compact difference schemes for the approximation of an arbitrary order derivative. We formulate a methodology for calculating the approximation coefficients involving a large number of mesh points leading to very high-order formulas, e.g., 20th or 30th order. The main emphasis is on the first and second order derivative occurring in the Navier-Stokes equations governing fluid flow problems. Contrary to most studies on compact schemes in which the computational stencils employed in the discretization are narrow and consist of only a few mesh points K, in the presented paper we consider wide stencils that in practice are limited only by the overall number of the nodes in a computational grid. We analyze the features of such wide stencil based schemes in physical and spectral space. It is shown that the solutions obtained applying the schemes of the same order but involving different number of the mesh points characterize significantly different accuracy. This is presented by direct comparisons of the analytically derived formula for the leading terms of the truncation errors as well as in practical simulations. The test cases include a second order ordinary differential equation with variable coefficients, 1D Burgers equation and three typical benchmark problems used in Computational Fluid Dynamics (CFD), i.e., inviscid vortex advection, convected Taylor-Green vortices and a double jet flow. The obtained results are compared with available analytical solutions and the solutions obtained with the help of a pseudo-spectral method. We found that very high-order compact schemes reach the spectral accuracy characterized by the error decrease proportional to O(hK). In some cases, they show even higher accuracy, and moreover, reaching the limit of the machine precision they are substantially more resistant to the round-off errors.

3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes

PurposeThe purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry).Design/methodology/approachThis study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes.FindingsThe significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry).Originality/valueIt has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

Optimal local truncation error method for solution of elasticity problems for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes was recently developed for PDEs with homogeneous materials on regular and irregular domains as well as for the scalar time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. Here, OLTEM is extended to a system of time-independent elastic PDEs for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes. We show the development of OLTEM for the 2D elasticity equations using compact 9-point stencils that are similar to those for linear quadrilateral finite elements. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. There are no unknowns on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal second order of accuracy for OLTEM with 9-point stencils on unfitted Cartesian meshes. Numerical experiments for elastic heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom: a) OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; b) up to engineering accuracy of OLTEM with unfitted Cartesian meshes is even more computationally efficient than quadratic and cubic finite elements with much wider stencils and conformed meshes. The proposed technique yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems.

Two-scale methods for convex envelopes

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780].

Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Recently we have developed the optimal local truncation error method (OLTEM) for PDEs with constant coefficients on irregular domains and unfitted Cartesian meshes. However, many important engineering applications include domains with different material properties (e.g., different inclusions, multi-material structural components, etc.) for which this technique cannot be directly applied. In the paper OLTEM is extended to a much more general case of PDEs with discontinuous coefficients and can treat the above-mentioned applications. We show the development of OLTEM for the 1-D and 2-D scalar wave equation as well as the heat equation using compact 3-point (in the 1-D case) and 9-point (in the 2-D case) stencils that are similar to those for linear quadrilateral finite elements. Trivial unfitted Cartesian meshes are used for OLTEM with complex interfaces between different materials. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. The calculation of the unknown stencil coefficients 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 a given width of stencil equations. In contrast to the second order of accuracy for linear finite elements, OLTEM provides the fourth order of accuracy in the 1-D case and in the 2-D case for horizontal interfaces as well as the third order of accuracy for the general geometry of smooth interfaces. The numerical results for the domains with complex smooth interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than quadratic finite elements and yields the results close to those for cubic finite elements with much wider stencils. The wave and heat equations are uniformly treated with OLTEM. OLTEM can be directly applied to other partial differential equations.

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29,30]. Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14,29,30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.

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.

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.

The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

Recently we have developed a new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Poisson equation on irregular domains with the Dirichlet boundary conditions. Its extension to the Neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used with the new approach for 3-D irregular domains. The Neumann boundary conditions are introduced as the known right-hand side into the stencil and do not change the width of the stencil. 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. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-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 fourth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

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.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

A new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains with the Dirichlet boundary conditions has been developed. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used for 3-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 of the new technique. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-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 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 fifth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

Compact high-order numerical schemes for scalar hyperbolic partial differential equations

Abstract This work is a comparative study on the numerical behavior of compact high-order schemes for discretizing scalar hyperbolic problems. To this aim, we investigate two possibilities: a high-order Hermitian scheme (HUPS), based upon the scalar variable and its first derivatives in space, and a “classical” high-order upwind scheme (CUPS), based upon the scalar variable and a wide numerical stencil, as a reference scheme. To study the theoretical properties of both schemes we use a spectral analysis based on a Fourier transform of the numerical solution. In a one-dimensional context, this analysis enables us to investigate the impact of the “spurious” components generated by the Hermitian schemes. As demonstrated by the spectral analysis, in some circumstances, this “spurious” component may deeply alter the accuracy of the scheme and even, in some cases, destroy its consistency. Thus, by identifying the salient features of a Hermitian scheme, we try to design efficients Hermitian schemes for multi-dimensional purposes. A two-dimensional Fourier analysis and some scalar numerical tests allow to highlight two high-order versions of a Hermitian scheme (HUPS4,5) of which the numerical properties are compared with a classical fifth-order scheme (CUPS5) based upon a wider stencil.