High-order Schemes Research Articles

Discrete variance decay analysis of spurious mixing

This work examines the use of discrete variance decay of tracers to estimate locally in space and time the numerical mixing caused by different processes during a tracer transport step. Expressions for local discrete variance decay (DVD) rates are directly derived from discrete tracer equations without any assumptions on discrete fluxes of the second moment. They relate the DVD rates to the fluxes of the first moment through the faces of scalar control cell. Mixing associated with advective and diffusive fluxes is thus estimated. The new framework avoids the need for second-moment flux definition when solved directly on finite-volume cell faces but still invokes certain second-moment fluxes when the face DVD rates are partitioned to cells sharing the face. These implied discrete fluxes depend on the partitioning and are non-unique. For third- or higher-order advection schemes, the DVD rates are contaminated by dispersive errors intrinsic to the approach, introducing uncertainty to the locality of any estimates produced by it. Additional temporal averaging or coarse-graining is thus necessary. Through the application of this technique, Numerical mixing is found to be correlated with the distribution of eddy kinetic energy. Numerical mixing induced by vertical advection is found to be relatively small and correlated with the distribution of buoyancy fluxes. The explored high-order schemes are found to demonstrate levels of spurious mixing which may locally exceed physical mixing.

A New High-Order Fractional Parallel Iterative Scheme for Solving Nonlinear Equations

Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method’s stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method’s efficiency, stability, and consistency compared to existing methods.

Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes

Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes

Simulation of a Mesoscale Convective System Using a New Formulation for Advection in a Bin Microphysics: Sensitivity to Advection, Microphysics, and Grid Spacing

Abstract A computationally efficient semi-Lagrangian advection (SLA) scheme designed for microphysical variable advection was implemented into the Weather Research and Forecasting (WRF) Model with spectral bin microphysics (SBM). The primary goal was to reduce the CPU time for advection of the scalar SBM variables and demonstrate its applicability to simulation of a real-data case study in the eta vertical coordinate system. A mesoscale convective system (MCS) in Midlatitude Continental Convective Clouds Experiment (MC3E-0520) was selected for simulations. We compared the SLA and high-order, nonlinear monotonic advection schemes and tested the sensitivity of the simulated radar reflectivity, microphysical, and dynamic properties of the MCS to the choice of microphysical schemes, aerosol concentration, and grid spacing. Simulations using the SLA and monotonic advection schemes were similar, and the differences between them were much smaller than those between the SBM and bulk microphysical schemes tested. Improvement of the grid resolution had a larger impact on the vertical velocity field than did the choice of aerosol concentration. The total computational time in simulations with SLA was about 25% shorter than that with monotonic advection, which resulted from a reduction of more than 50% in computational time required for advection of the microphysical variables. Significance Statement Clouds comprise particles of various sizes and types: aerosols, liquid droplets, ice crystals, hail, and snow. Two major approaches describe the time and spatial evolution of these particle species: solving basic microphysical (bin microphysics) and simplified parameterization (bulk microphysics) equations. Bin microphysics is time-consuming as it operates with size distributions described by multiple mass bins. Among the microphysical processes, the advection of the microphysical variables requires significant CPU time. We tested a new numerical advection scheme that requires several times less computer time than the standard high-order nonlinear schemes being of similar accuracy. Due to the introduced change, the advection of droplet size distributions became less of a factor affecting the total computer time of bin-type schemes. This is an important step toward making the bin approach computationally efficient without losing accuracy. It is shown that the sensitivity of the results to spatial grid spacing and to the choice of microphysical schemes is significant and may be larger than that to variations in the aerosol concentration.

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

Enhancing performance of end-to-end communication system using Attention Mechanism-based Sparse Autoencoder over Rayleigh fading channel

Deep learning has revolutionized communication systems by introducing innovative approaches to address channel impairments through end-to-end models. Autoencoders, a type of deep learning architecture, are adept at learning compact data representations. However, conventional autoencoders in end-to-end models can suffer from overfitting, which limits their effectiveness in noisy communication environments. To address this issue, we propose a Sparse Autoencoder-based (SAE) model that enforces sparsity and promotes the extraction of robust features. Despite its effectiveness, the SAE model may still lack the ability to focus on the most relevant features of the input data. To overcome this limitation, we further introduce an Attention Mechanism-based Sparse Autoencoder (ASA) model. This model integrates the feature extraction capabilities of a sparse autoencoder with an attention mechanism that selectively highlights informative features of the signal. Through simulations, we demonstrate that both proposed models significantly improve M-PSK and M-QAM communication system performance. When trained at 7 dB, both proposed models exhibit significant performance improvements at higher testing average SNRs. Our results show that the SAE model outperforms the conventional Maximum Likelihood Detection (MLD) model and baseline autoencoder systems but suffers from error floor issues. The SAE model suffers from an error floor at average SNRs beyond 16 dB for BPSK and 14 dB for higher-order modulation schemes. As the value of M increases, the performance gap between the MLD and the proposed SAE model narrows. The ASA model, however, effectively mitigates the error floor observed in the SAE model for all values of M and across all modulation schemes. This research highlights the benefits of integrating an attention mechanism with SAE, resulting in enhanced robustness and reliability in communication systems characterized by improved accuracy and reduced error rates.

Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method

The adaptive implicit method (AIM) is the mainstream approach for compositional reservoir simulation. In its standard form, AIM uses single-point upwinding to reconstruct the fluxes across the interfaces of the control volume, and this leads to substantial numerical diffusion and loss of accuracy. Previous efforts to improve the accuracy of AIM focused on using high-order schemes to reconstruct the interfacial fluxes; those schemes introduced additional numerical nonlinearities to the system of equations or compromised the accuracy of the Jacobian by neglecting the high-order terms in its construction. In this work, we describe a high-resolution compact-stencil (HRCS) AIM. The new scheme is applied to compositional reservoir simulation. In addition to the mixed implicit/explicit time discretization of standard AIM, the HRCS AIM scheme uses a mixed time and space discretization. Specifically, we blend low- and high-order fluxes according to a well defined rule that uses a high-order reconstruction in the explicit regions of the domain and a low-order reconstruction in the implicit regions. This strategy ensures that additional nonlinearities introduced by the high-order reconstruction do not impact the Jacobian matrix, thus preserving the algebraic structure of standard AIM. The HRCS AIM method is demonstrated using several compositional problems. The results indicate substantial gains in accuracy with a small additional computational cost compared with standard AIM. Additionally, HRCS AIM is more robust and has a smaller computational cost compared with its full high-resolution counterpart.

Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge–Kutta discontinuous Galerkin method

The Runge–Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the P k P^k approximation in the final stage, even if we drop the k k th-order polynomial modes and use the P k − 1 P^{k-1} approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed by Chen, Sun, and Xing [arXiv preprint, arXiv:2402.15150, 2024]. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability type and accuracy order of the original method.

Numerical study on the combustion characteristics and performances of single and multi-injectors in a scramjet combustor

The present study numerically investigates the combustion characteristics and performance of a direct-connect gaseous hydrogen-fueled scramjet combustor depending on the injector scheme. A comprehensive numerical simulation was conducted with an improved delayed detached eddy simulation (IDDES) approach. The framework utilized a high-order accurate numerical scheme to ensure the high fidelity of the results. A total of ten cases were considered combining two injector schemes and five injection pressure conditions. Each injector scheme had a similar range of global equivalence ratios. Numerical results revealed the differences in the local dynamics of the counter-rotating vortex pair. The multi-injector case did not maintain the jet's systemic vortex structure, which plays a primary role in the fuel-air mixing and burning. It owes to the interactions between the jet-jet and the jet-wall surface, where the interaction leads to the loss of momentum. This characteristic of the multi-injector makes the fuel-air mixing contact surface get closer to a thin-flat layer, resulting in the flame being anchored on a flat shear layer over the entire combustor. As a result, the combustion efficiency of the multi-injector is much lower than that of the single injector under a similar equivalence ratio range. Present results indicate that the multi-injector, which is expected to increase the combustion performance by maximizing the fuel-air contact surface, may operate in contrast to its original anticipation under certain configurations and conditions. It also suggests that optimizing the combustion performance requires careful design of injector distributions considering the distances and interactions between injector-to-injector and injector-to-wall.

An energy stable high-order cut cell discontinuous Galerkin method with state redistribution for wave propagation

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to “cut” a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani in [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is L2 energy stable under arbitrary quadrature. We prove that state redistribution can be added to a provably L2 energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's L2 stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

Implicit high-order gas-kinetic schemes for compressible flows on three-dimensional unstructured meshes II: Unsteady flows

For the simulations of unsteady flow, the global time step becomes really small with a large variation of local cell size. In this paper, an implicit high-order gas-kinetic scheme (HGKS) is developed to alleviate the restrictions on the time step for unsteady simulations. In order to improve the efficiency and keep the high-order accuracy, a two-stage third-order implicit time-accurate discretization is proposed. In each stage, an artificial steady solution is obtained for the implicit system with the pseudo-time iteration. In the iteration, the classical implicit methods are adopted to solve the nonlinear system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and generalized minimum residual (GMRES) methods. To achieve the spatial accuracy, the HGKSs with both non-compact and compact reconstructions are constructed. For the non-compact scheme, the weighted essentially non-oscillatory (WENO) reconstruction is used. For the compact one, the Hermite WENO (HWENO) reconstruction is adopted due to the updates of both cell-averaged flow variables and their derivatives. The expected third-order temporal accuracy is achieved with the two-stage temporal discretization. For the smooth flow, only a single artificial iteration is needed. For uniform meshes, the efficiency of the current implicit method improves significantly in comparison with the explicit one. For the flow with discontinuities, compared with the well-known Crank-Nicholson method, the spurious oscillations in the current schemes are well suppressed. The increase of the artificial iteration steps introduces extra reconstructions associating with a reduction of the computational efficiency. Overall, the current implicit method leads to an improvement in efficiency over the explicit one in the cases with a large variation of mesh size. Meanwhile, for the cases with strong discontinuities on a uniform mesh, the efficiency of the current method is comparable with that of the explicit scheme.

A unified approach to the construction of higher-order derivative-free iterative methods for solving systems of nonlinear equations

In this article, we introduce a unified approach to constructing a higher-order derivative-free scheme based on the approximations of F'(zk)-1. A family of order p=6,7 derivative-free method is proposed and compared to some well-known methods. The necessary and sufficient condition for p-th order of convergence are given in terms of parameter matrices τ(k) and α(k) . Some good choices of and are offered. Numerical experiments were carried out to confirm the theoretical results.

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

On the Use of Different Sets of Variables for Solving Unsteady Inviscid Flows with an Implicit Discontinuous Galerkin Method

This article presents a comparison between the performance obtained by using a spatial discretization of the Euler equations based on a high-order discontinuous Galerkin (dG) method and different sets of variables. The sets of variables investigated are as follows: (1) conservative variables; (2) primitive variables based on pressure and temperature; (3) primitive variables based on the logarithms of pressure and temperature. The solution is advanced in time by using a linearly implicit high-order Rosenbrock-type scheme. The results obtained using the different sets are assessed across several canonical unsteady test cases, focusing on the accuracy, conservation properties and robustness of each discretization. In order to cover a wide range of physical flow conditions, the test-cases considered here are (1) the isentropic vortex convection, (2) the Kelvin–Helmholtz instability and (3) the Richtmyer–Meshkov instability.

Comparative study on predicting turbulent kinetic energy budget using high-order upwind scheme and non-dissipative central scheme

The accurate computation of different turbulent statistics poses different requirements on numerical methods. In this paper, we investigate the capabilities of two representative numerical schemes in predicting mean velocities, Reynolds stress and budget of turbulent kinetic energy (TKE) in low Mach number flows. With concerns on numerical order of accuracy, dissipation and dispersion properties, a high-order upwind scheme with relatively good dispersion and dissipation and a second-order non-dissipative central scheme with perfect dissipation but poor dispersion are adopted for this comparative study. By carrying out a series of numerical simulations including Taylor-Green vortex, turbulent channel flow at Reτ=180\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ au }=180$$\\end{document} and turbulent flow over periodic hill at Reb=10595\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{b}=10595$$\\end{document}, it can be obtained that although the high-order upwind scheme lacks perfection of dissipation in high wave number range, it still demonstrates superior predictive capability compared with the second-order non-dissipative central scheme, especially with relatively coarse grids. Finally, by taking the high-order upwind scheme as a suitable selection for turbulence simulation, the turbulent flow over a 30P30N multi-element airfoil is investigated as an application study. After briefly comparing the simulated profiles and spectrum with reference experimental results as validation, the budget of TKE is analyzed to locate the dominant flow structures and regions. It is found that the production and dissipation terms behave in a “monopole” pattern in the locations with strong shears and wakes. Whereas the advection and diffusion terms show an “inward” pattern and an “outward” pattern, which indicate the spatial transport of TKE between the center of the shear layer and nearby locations.

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

High-order accurate implicit-explicit time-stepping schemes for wave equations on overset grids

New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids. The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations.

Improving approximation accuracy in Godunov-type smoothed particle hydrodynamics methods

The study examines the origin of errors resulting from the approximation of the right hand sides of the Euler equations using the Godunov type contact method of smoothed particle hydrodynamics (CSPH). The analytical expression for the numerical shear viscosity in CSPH method is obtained. In our recent study the numerical viscosity was determined by comparing the numerical solution of momentum diffusion in the shear flow with theoretical one. In this study we deduce the analytical expression for the numerical viscosity which is found to be similar to numerical one, confirming the obtained results. To reduce numerical diffusion, diffusion limiters are typically applied to expressions for contact values of velocity and pressure, as well as higher-order reconstruction schemes. Based on the performed theoretical analysis, we propose a new method for correcting quantities at interparticle contacts in CSPH method, which can be easily extended to the MUSCL-type (Monotonic Upstream-centered Scheme for Conservation Laws) method. Original CSPH and MUSCL-SPH approaches and ones with aforementioned correction are compared.

The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, L2 stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.