Third-order Scheme Research Articles

A note of caution on numerical scheme selection: Evidence from cyclone separator CFD simulations with appropriate near-wall grid sizes

In this paper two things are done. (i) The appropriate numerical scheme set for computational fluid dynamics (CFD) simulations of cyclone separators with various vortex-finder-to-cone-tip diameter ratios (D⁎) has been achieved by comparing the mean flow patterns and performances simulated by two previously proposed numerical scheme sets. The predicted results revealed that the CFD simulations of different cyclone separator designs require the third-order accuracy scheme and proper near-wall grid sizes to preserve flow similarity, especially for cyclone separators with D⁎ < 1. Therefore, the present research alerts caution concerning the lower-order numerical scheme for cyclone separator CFD simulation. (ii) The present CFD work confirms that the near-wall grid size estimation method originally developed for gas cyclones is a possible method for estimating near-wall grid sizes for hydrocyclone with air core diameter (Da) ≤ 0.4Dv. In addition, the capabilities of pressure-strain sub-models for cyclone separator CFD simulations were preliminarily discussed.

Two families of second-order fractional numerical formulas and applications to fractional differential equations

Two families of second-order difference formulas called fractional BT- $$\theta $$ and fractional BN- $$\theta $$ for the fractional calculus are proposed by introducing a free parameter $$\theta $$ aiming to generalize the classic fractional BDF2, the fractional trapezoidal rule (FTR) and the second-order generalized Newton-Gregory formula (GNGF2). The error bounds for these two families of novel formulas are analysed indicating that fractional BT- $$\frac{1}{2}$$ (or FTR) is superior to other formulas for integral cases. At least two advantages can be observed for the introduction of $$\theta $$ : (i) The fractional BN- $$\theta $$ exhibits superconvergence for an appropriate $$\theta $$ which permits us to develop more robust third-order schemes for problems such as the diffusion-wave equation, compared with the fractional BDF3. (ii) The fractional BT- $$\theta $$ allows us to take the advantage of FTR that a sub-optimal error bound can be obtained (by taking $$\theta $$ close to $$\frac{1}{2}$$ ) when directly discretizing fractional derivatives. Further, some correction techniques are discussed to overcome the solution initial singularity in numerical tests. The numerical results confirm the correctness of the theoretical analysis and the efficiency of our scheme.

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

To improve the shock-capturing capability of the third-order WENO scheme and enhance its computational efficiency, in this paper, we design a new WENO scheme independent of the local smoothing factor, WENO-SIF. The weight function of the WENO-SIF scheme is the segmentation function of the sub-stencil, which is guaranteed to achieve the desired accuracy at higher order critical points. WENO-SIF does not need to compute the smoothing factor during the computation, which effectively reduces the computational consumption. The present WENO-SIF is compared with WENO-JS and other WENO schemes for numerical experiments at one- and two-dimensional benchmark problems with a suitable choice of $$\lambda =0.13$$ . The results demonstrate that the WENO scheme can further improve the resolution of WENO-JS, achieve optimal accuracy at high-order critical points, and significantly reduce the computational consumption.

Unconditionally Stable Numerical Scheme for Heat Transfer of Mixed Convective Darcy–Forchheimer Flow of Micropolar Fluid Over Oscillatory Moving Sheet

AbstractThis contribution proposes a third-order numerical scheme for solving time-dependent partial differential equations (PDEs). This third-order scheme is further modified, and the new scheme is obtained with second-order accuracy in time and is unconditionally stable. The unconditional stability of the new scheme is proved by employing von Neumann stability analysis. For spatial discretization, a compact fourth-order accurate scheme is adopted. Moreover, a mathematical model for heat transfer of Darcy–Forchheimer flow of micropolar fluid is modified with an oscillatory sheet, nonlinear mixed convection, thermal radiation, and viscous dissipation. Later on, the dimensionless model is solved by the proposed second-order scheme. The results show that velocity and angular velocity have dual behaviors by incrementing coupling parameters. The proposed second-order accurate in-time scheme is compared with an existing Crank–Nicolson scheme and backward in-time and central in space (BTCS) scheme. The proposed scheme is shown to have faster convergence than the existing Crank–Nicolson scheme with the same order of accuracy in time and space. Also, the proposed scheme produces better order of convergence than an existing Crank–Nicolson scheme.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

In this paper, we propose four time marching schemes for the two-mode phase field crystal equation with the periodic boundary condition. The first- and second-order schemes are based on the stabilized semi-implicit approach with a combination of the backward Euler method, Crank–Nicolson method, and second-order backward differentiation formula, respectively. Based on the convex splitting method and third-order backward differentiation formula, a third-order scheme is developed. Different stabilized terms are added so that the energy stability of the proposed schemes can be guaranteed. Theoretically, the unique solvability and energy stability of the proposed schemes are rigorously proved. The error estimate of the first-order scheme is also given. Various numerical examples and simulations in the 2D and 3D cases are presented to demonstrate the accuracy, stability, and efficiency of the proposed schemes.

Finite difference schemes for MHD mixed convective Darcy–forchheimer flow of Non-Newtonian fluid over oscillatory sheet: A computational study

This contribution proposes two third-order numerical schemes for solving time-dependent linear and non-linear partial differential equations (PDEs). For spatial discretization, a compact fourth-order scheme is deliberated. The stability of the proposed scheme is set for scalar partial differential equation, whereas its convergence is specified for a system of parabolic equations. The scheme is applied to linear scalar partial differential equation and non-linear systems of time-dependent partial differential equations. The non-linear system comprises a set of governing equations for the heat and mass transfer of magnetohydrodynamics (MHD) mixed convective Casson nanofluid flow across the oscillatory sheet with the Darcy–Forchheimer model, joule heating, viscous dissipation, and chemical reaction. It is noted that the concentration profile is escalated by mounting the thermophoresis parameter. Also, the proposed scheme converges faster than the existing Crank-Nicolson scheme. The findings that were provided in this study have the potential to serve as a helpful guide for investigations into fluid flow in closed-off industrial settings in the future.

A Hybrid Five-Level Modular Multilevel Converter With High Efficiency and Small Energy Storage Requirements for HVDC Transmission

This article presents a hybrid five-level converter (H5LC) with cascaded full-bridge submodules (FBSMs) for high voltage direct current (HVdc) transmission. A five-level converter operates at a fundamental switching frequency to generate symmetric five-level square wave output voltages which are shaped into smooth sinusoidal voltages by the ac-side FBSMs. Using a proposed third-order harmonic voltage injection scheme, the FBSMs’ total blocking voltage is limited to one-eighth of the dc-side voltage, reducing the H5LC's losses, footprint, and capital costs. dc pole-to-pole fault blocking is enabled using a bypass branch consisting of bidirectional thyristor valves and fast mechanical switches to suppress the ac-side fault contribution within half a fundamental cycle. Full-scale HVdc simulation studies show the H5LC's ability to control real and reactive powers and additionally, its dc fault resilience. A lab-scale hardware implementation of an H5LC with 30 FBSMs is presented to verify its operating principle. In contrast to the half-bridge submodule based modular multilevel converter, the H5LC has improved efficiency, fewer semiconductor devices, and significantly reduced energy storage requirements.

Efficient third-order BDF finite difference scheme for the generalized viscous Burgers’ equation

This article presents a third-order backward differentiation formula (BDF3) finite difference scheme for the generalized viscous Burgers’ equation. The discretization of time and space directions is accomplished by the BDF3 method and standard second-order difference formula, respectively, thereby constructing a fully-discrete scheme. For the proposed scheme, we yield the convergence of h2+τ3 by means of the energy argument and the cut-off function method. Besides, a comparison of the time convergence rate and numerical accuracy with those of recent existing work shows the effectiveness and competitiveness of our approach. A numerical experiment is carried out to verify the theoretical predictions.

A Two-Dimensional Third-Order CESE Scheme for Ideal MHD Equations

A Two-Dimensional Third-Order CESE Scheme for Ideal MHD Equations

A numerical scheme for Darcy-Forchheimer flow of non-Newtonian nanofluid under the effects of convective and zero mass flux boundary conditions

This research aims to propose a numerical scheme for solving boundary value problems. It is a two-stage, third-order accurate scheme known as a predictorcorrector scheme. The two main results are finding the region of the scheme where it is stable and determining the stability criterion for a set of linearized first-order differential equations. In addition, a mathematical model for heat and mass transfer of Darcy-Forchheimer flow of non-Newtonian nanofluid over the sheet is presented. The similarity transformations reduce PDE into a system of ODE for easier manipulation. The results are compared with the past research and those obtained by MATLAB SOLVER BVP4C. The results show that the velocity profile slightly decays by enhancing the Weisenberg number.

Resolving turbulent boundary layer on coarse grid using function enrichment based on variational reconstructions

An improved finite volume scheme based on variational reconstruction and function enrichment has been proposed in this paper. By incorporating the law-of-the-wall into the variational reconstruction, the proposed method can resolve turbulent flow accurately on grids much coarser than those needed by traditional methods. The usual reconstruction in a finite volume scheme assumes that the solution is belonging to a polynomial function space, which is inaccurate to resolve the velocity profile within the turbulent boundary layer unless the grid in wall-normal direction is fine enough. In the present paper, this function space is “enriched” by adding a basis function that is derived from the logarithmic law of the turbulent boundary layer. Then variational reconstruction procedure is applied to find the “best” solution belonging to the expanded function space. The advantage of the present method over the traditional wall function model is that the turbulent flow within the boundary layer is resolved rather than modeled. The algorithms and the implementations are discussed in detail. The proposed method is applied to the turbulent flow over a flat plate at a Reynolds number of 5×106 and the turbulent flow over a plate with a bump at a Reynolds number of 3×106. The results of second- and third-order schemes are presented for the turbulent velocity profile and the skin friction coefficients. The numerical results suggest that this approach not only resolves the near wall turbulent accurately on very coarse grids but also reduces the computational time significantly.

Computation of Two-Dimensional Poisson Equation Using the Third-Order Discrete Scheme of Finite Difference Method Based on Node Set Vector

A novel third-order discrete scheme of finite difference method based on node set vector for two dimensional Poisson equation is proposed in this paper. Studies on the basic discrete scheme of this method as well as the discrete scheme of the interior node and boundary node are carried out in detail. Computer programs are also developed to emulate a computation sample. The numerical computation results show that this discrete scheme is effective for the numerical computation of two dimensional Poisson equation.

A Faster Third-Order Masking of Lookup Tables

Masking of S-boxes using lookup tables is an effective countermeasure to thwart side-channel attacks on block ciphers implemented in software. At first and second orders, the Table-based Masking (TBM) schemes can be very efficient and even faster than circuit-based masking schemes. Ever since the customised second-order TBM schemes were proposed, the focus has been on designing and optimising Higher-Order Table-based Masking (HO-TBM) schemes that facilitate masking at arbitrary order. One of the reasons for this trend is that at large orders HO-TBM schemes are significantly slower and consume a prohibitive amount of RAM memory compared to circuit-based masking schemes such as bit-sliced masking, and hence efforts were targeted in this direction. However, a recent work due to Valiveti and Vivek (TCHES 2021) has demonstrated that the HO-TBM scheme of Coron et al. (TCHES 2018) is feasible to be implemented on memory-constrained devices with pre-processing capability and a competitive online execution time. Yet, currently, there are no customised designs for third-order TBM that are more efficient than instantiating a HO-TBM scheme at third order.In this work, we propose a third-order TBM scheme for arbitrary S-boxes that is secure in the probing model and under compositions, i.e., 3-SNI secure. It is very efficient in terms of the overall running time, compared to the third-order instantiations of state-of-the-art HO-TBM schemes. It also supports the pre-processing functionality. For example, the overall running time of a single execution of the third-order masked AES-128 on a 32-bit ARM-Cortex M4 micro-controller is reduced by about 80% without any overhead on the online execution time. This implies that the online execution time of the proposed scheme is approximately eight times faster than the bit-sliced masked implementation at third order, and it is comparable to the recent scheme of Wang et al. (TCHES 2022) that makes use of reuse of shares. We also present the implementation results for the third-order masked PRESENT cipher. Our work suggests that there is a significant scope for tuning the performance of HO-TBM schemes at lower orders.

A New Third-Order Finite Difference WENO Scheme to Improve Convergence Rate at Critical Points

In this work, a new, improved third-order finite difference weighted essentially non-oscillatory scheme is presented for one- and two-dimensional hyperbolic conservation laws and associated problems. The parameter p which is regulate dissipation is introduced in the nonlinear weights in the framework of the conventional WENO-Z scheme, and the higher-order global smoothness indicator is obtained by the idea of Wang [Wang, Y. H., Y. L. Du, K. L. Zhao and L. Yuan. 2020. ‘A Low-dissipation Third-order Weighted Essentially Nonoscillatory Scheme with a New Reference Smoothness Indicator’. International Journal for Numerical Methods in Fluids. 92 (9): 1212–1234.], the sufficient condition of nonlinear weights is proved by using Taylor expansions. Finally, the value range of parameter p is obtained. The proposed scheme is verified to achieve the optimal order near critical points by linear convection equations with different initial values, and the high-resolution characteristic of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.

A third-order weighted nonlinear scheme for hyperbolic conservation laws with inverse Lax-Wendroff boundary treatment

Cartesian grids are often used in applications due to the simplicity of grid generation and the efficiency of discretization algorithms. However, for problems with curved boundaries, grid points are often away from the boundaries, leading to the issue of imposing boundary conditions. Recently, it was shown that the inverse Lax-Wendroff (ILW) boundary treatment is very useful for addressing this issue. While most of the investigations focus on the WENO schemes for hyperbolic conservation laws, we present in this paper a new third-order weighted nonlinear scheme, which is based on the framework of weighted compact nonlinear schemes. We show that the scheme is applicable directly on Cartesian grids by using the ILW boundary treatment. To demonstrate the linear stability of the scheme, both the method of eigenvalue analysis and the Gustafsson-Kreiss-Sundström (GKS) theory are employed and analyzed in details. Some numerical tests are also performed to show the accuracy and validity of the proposed method.

Acceleration and performance analysis of a compressible Euler solver with CUDA

To develop high performance computing methods for compressible flow calculation, a GPU-accelerated compressible flow solver is developed with Compute Unified Device Architecture (CUDA). The WENO5 scheme is adopted for spatial discretization, and the third-order Runge-Kutta scheme is used for time discretization. According to the algorithm and programming model, the heterogeneous computing method of the solver is designed. Different kernels are designed to implement different computing functions, and shared memory is used for time-advanced computations. The solver is verified by the one-dimensional shock tube case, and a good acceleration effect is obtained with the increase of the grid size. And the impact of execution configuration on kernel performance was investigated. When the block size is reduced under different grid sizes, the speedup changes in the same way, but the performance parameters change differently.

A less time-consuming upwind compact difference method with adjusted dissipation property for solving the unsteady incompressible Navier-Stokes equations

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20] .

Third-order scale-independent WENO-Z scheme to achieve optimal order at critical points

Our past work has shown that when the critical points occur within grid intervals, the relations of accuracy of the smoothness indicators of weighted essentially non-oscillatory schemes (WENO) reported by Jiang and Shu differ from those that are obtained by assuming that the critical points occur on the grid nodes. The global smoothness indicator in the WENO-Z scheme might accordingly differ from the original one. We use this understanding to first discuss several issues regarding current improvements to third-order WENO-Z (e.g., WENO-NP3, -F3, -NN3, and -PZ3), i.e., numerical results with scale dependence, the validity of the analysis by assuming that the critical points occur on the nodes, and sensitivity in terms of the computational time step and initial conditions through the examination of the order of convergence. Numerical simulations and analyses were used to highlight defects in these improvements that occur either due to the scale dependence of the results, or the failure to recover the optimal order when the critical points occur on the half-nodes. Following this, a generic analysis that assumes that the first-order critical points occur within the grid intervals is provided. The theoretical results thus derived are used to propose two scale-independent third-order WENO-Z schemes that can be used to attain the optimal order at the critical points. The first scheme is obtained by extending a downstream smoothness indicator to derive a new global smoothness indicator and incorporating it into the mapping function. The second scheme is achieved by extending another smoothness indicator and using a different global indicator. The following validations are chosen and tested: the typical 1D problem of scalar advections, and 1D and 2D problems based on Euler's equations. The results verify the capability of the proposed schemes to recover the optimal order at the critical points. Moreover, the first of the above two proposed schemes outperforms the improved third-order WENO-Z scheme in terms of numerical resolution and robustness, which is usually favored by applications.