High-order Numerical Schemes Research Articles

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.

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.

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.

Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows

The scalar auxiliary variable (SAV) approach was considered by Shen et al. in Shen et al. (2019) and has been widely used to simulate a series of gradient flows. However, the SAV-based schemes are known for the stability of a ‘modified’ energy. In this paper, we construct a series of modified SAV approaches with unconditional energy dissipation law based on several improvements to the classic SAV approach. Firstly, by introducing the three-step technique, we can reduce the number of constant coefficient linear equations that need to be solved at each time step, while retaining all of its other advantages. Secondly, the addition of energy-optimal technique and Lagrange multiplier technique can make the numerical schemes have the advantage of preserving the original energy dissipation. Thirdly, we use the first-order approximation of the energy balance equation in the GSAV approach, instead of discretizing the dynamic equation of the auxiliary variable, so that we can construct the high-order unconditional original energy stable numerical schemes. Finally, representative numerical examples show that the efficiency and accuracy of the proposed schemes are improved.

Design, Fabrication, and Commissioning of Transonic Linear Cascade for Micro-Shock Wave Analysis

Understanding shock wave behavior in supersonic flow environments is critical for optimizing the aerodynamic performance of turbomachinery components. This study introduces a novel transonic linear cascade design, focusing on advanced blade manufacturing and experimental validation. Blades were 3D-printed using Inconel 625, enabling tight control over the geometry and surface quality, which were verified through extensive dimensional accuracy assessments and surface finish quality checks using coordinate measuring machines (CMMs). Numerical simulations were performed using Ansys CFX with an implicit pressure-based solver and high-order numerical schemes to accurately model the shock wave phenomena. To validate the simulations, experimental tests were conducted using Schlieren visualization, ensuring high fidelity in capturing the shock wave dynamics. A custom-designed test rig was commissioned to replicate the specific requirements of the cascade, enabling stable and repeatable testing conditions. Experiments were conducted at three different inlet pressures (0.7-bar, 0.8-bar, and 0.9-bar gauges) at a constant temperature of 21 °C. Results indicated that the shock wave intensity and position are highly sensitive to the inlet pressure, with higher pressures producing more intense and extensive shock waves. While the numerical simulations aligned broadly with the experimental observations, discrepancies at finer flow scales suggest the need for the further refinement of the computational models to capture detailed flow phenomena accurately.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

Best-estimate water hammer simulations to avoid the calculation of unrealistically high loads or unphysical pressure and force peaks

For safety-related systems fluid loads due to fluid transients have to be quantified for subsequent structural analyses to ensure their integrity or function, as required. Usually transient fluid loads in pipe systems are determined with one-dimensional water hammer software. For single-phase liquid flow, the method of characteristics (MOC) is often used that gives in this case appropriate results. For the consideration of local vapor bubbles, the MOC is combined with the discrete vapor cavity model (DVC). The DVC model may generate unrealistic pressure spikes due to the calculation of the collapse of multi-cavities in scenarios, where only one vapor bubble should actually occur. The application of a two-phase code may improve the calculation results. One requirement for the latter codes is the ability to calculate the propagation of steep gradients without suffering from numerical diffusion to exclude the underestimation of fluid loads. This is commonly attained by applying higher-order numerical schemes. However, the application of a numerical method of pure 2nd order leads to the calculation of unphysical oscillations at steep gradients causing severe problems during the solution. To exclude this, numerical methods with flux limiters can be used. With their application, the calculation of unrealistically high loads due to numerical deficiencies can be minimized. In addition, the consideration of further physical effects, that lead to the reduction of loads during transient flow processes, allows for a more realistic calculation of the loads. These are unsteady friction, widening of the pipe caused by pressure increase, fluid-structure interaction at junctions and due to friction, degassing of gas that is initially dissolved physically in a liquid and thermodynamic non-equilibrium during vapor bubble collapse. The in-house code DYVRO applies a second-order accurate scheme with flux limiters based on the Godunov method and can account for the above-described physical phenomena. It is shown by comparison of calculation results obtained by DYVRO with experimental data from literature that with modeling of these physical effects the loads can be calculated more realistically. Generally, these loads are lower than the results calculated by simplified models, which do not account for these effects. Considering that these loads are applied in subsequent structural analyses, cost-intensive oversizing of pipes and their supports can be avoided, by ensuring the necessary safety.

Maximum bound principle preserving additive partitioned Runge-Kutta schemes for the Allen-Cahn equation

In this paper, a novel framework for constructing the temporal high-order numerical schemes that inherit the maximum bound principle (MBP) of the Allen-Cahn equation is proposed. The original Allen-Cahn equation is converted into an equivalent system by introducing an auxiliary variable. Based on the new system, we put forward and analyze high-order (up to the fourth-order) additive partitioned Runge-Kutta (APRK) schemes for the time integration of the Allen-Cahn equation, which satisfies the discrete MBP under reasonable time step constraints. A simple sufficient condition is also given to ensure that a class of APRK methods preserves the discrete MBP and that the resulting scheme is linearly implicit. The convergence order in the discrete L∞-norm is further provided by utilizing the established discrete MBP, which plays a vital role in the analysis. Several numerical experiments are carried out to validate our theoretical results.

Dynamic instability in lift-type reentry capsule at supersonic flow

Numerical simulations were conducted to investigate dynamic instability of a Japanese lift-type reentry capsule, which is named H-II transfer vehicle Recovery Vehicle (HRV), at Mach 1.2 by comparing two capsule shapes of an original HRV model and a model with a different shoulder angle of 10°. The 10° model has a thicker aft-body with a smaller shoulder angle than the original model. A previous pitch-direction One-Degree Of Freedom (1DOF) experiment revealed that the original model was dynamically stable, whereas the 10° model was dynamically unstable at Mach 1.2. So as to reproduce the same trends as the experiment, a high-order numerical scheme and zonal detached eddy simulation were employed. In fixed angle simulations, we found that the static longitudinal stability of the 10° model is higher than that of the original model in contrast with the observed dynamic stability. 1DOF free oscillation simulations successfully reproduced the same trends as the experiment. We elucidated that the main factor of the dynamic instability of the HRV-type capsule is a hysteresis of a boundary layer separation on the upper side of the capsule. The hysteresis is induced by a variation of momentum in a boundary layer due to capsule motion. We also found that for HRV-type capsules, the lower the static stability within the statically stable range, the higher the dynamic stability might be. In addition, on the lower side, a sign of the work exerted by surrounding flow is different between the original model and the 10° model. A dynamic mode decomposition analysis indicated that the vortex structures on the capsule wake might induce this difference.

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett et al. (2007) [8], due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable method, the highly efficient and accurate schemes (in time) with unconditional energy stability for solving the nonlocal Cahn-Hilliard equation are proposed. On the one hand, we have demonstrated the unconditional energy stability and mass conservation for the nonlocal Cahn-Hilliard equation with its high-order numerical schemes in the continuous and discrete level carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on fast Fourier transform and conjugate gradient approach for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.

A high order numerical scheme for a nonlinear nonlocal reaction–diffusion model arising in population theory

This paper provides a numerical method for nonlinear equation arising in mathematical biology. It is an extension of another one recently proposed for the linear, less realistic, situation. The main novel result is the proof that the convergence of the numerical method is of order four, as to our knowledge no similar high accuracy results exist yet in the current literature for usually employed simulation schemes for nonlocal equations.

A deep learning operator-based numerical scheme method for solving 1D wave equations

Abstract In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods.

High-order discretization–based self-adaptive turbulence eddy simulation for supersonic base flow with PHengLEI software

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

Enhanced numerical modeling of natural heat convective phase change for generalized non-Newtonian fluids at high Rayleigh number

Numerical modeling of convective heat transfer employing non-Newtonian fluids is a key issue and represent an interesting challenge in many engineering applications. For this reason, this work develops an enhanced numerical model to predict fast and accurately the convective phase change for generalized non-Newtonian fluids. The strongly thermally coupled model is solved by an improved pressure-correction algorithm (SIMPLERnP) and the Finite Volume Method with accuracy of the third order for the transient terms and second order for the convective boundary conditions and the velocity gradients that allows to calculate the apparent viscosity. The study examines the natural heat convective solidification of the ternary Al-27 %Cu-5.25 %Si alloy with the Herschel-Bulkley rheology at a high Rayleigh number (>107). In addressing the main problem, four crucial aspects are discussed: the validation of the numerical scheme with numerical and experimental results; the modeling of the non-linear temperature variation of the liquid phase change fraction; the non-Newtonian effects of the power index (0.1 ≤ n ≤ 1.9) and yield stress (0≤τ0≤5 Pa) on the heat transfer mechanisms; and finally, the time of computation with the high-order numerical scheme and the enhanced pressure-correction algorithm. The main finding is that FVM/SIMPLERnP provides a significant reduction in the time of computation compared to reliable pressure-correction schemes (SIMPLER and IDEAL), being a robust, accurate and efficient scheme to solve complex natural heat convective phase change problems involve generalized non-Newtonian fluids at high-Rayleigh numbers.

High Order Numerical Scheme for Generalized Fractional Diffusion Equations

High Order Numerical Scheme for Generalized Fractional Diffusion Equations