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In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution. [ABSTRACT FROM AUTHOR]

The deterministic methods are efficient for solving the neutron transport equation (NTE), but suffer from discretization errors. The increasingly complex geometric models make spatial discretization errors the primary source of discretization errors. Considering that spatial discretization errors come from inaccurate geometric shape descriptions and low-accuracy numerical schemes, this paper develops a Discontinuous Galerkin Finite Element Method for the NTE on unstructured polygonal meshes to reduce spatial discretization errors. In this method, the physical modal basis is adopted to handle the polygonal mesh and to achieve high-order accuracy in a uniform and efficient way. The numerical results of various fixed-source and k-eigenvalue benchmarks demonstrate that the method developed in this paper can give accurate solutions on polygonal meshes with high convergence rates. [ABSTRACT FROM AUTHOR]

It is challenging to apply numerical simulations to accurately predict the stall behavior of aircraft equipped with high-lift devices. Simulations with Reynolds-Averaged Navier Stokes (RANS) models suffer from lack of the reliability at high angles of attack with separated and reattached boundary layers, whereas wall-resolved Large Eddy Simulations (LES) of wall-bounded flows at high Reynolds numbers currently costs too much computational resources. A new unified hybrid turbulence modeling approach, denoted the Self-Adaptive Turbulence Eddy Simulation (SATES), is proposed and applied for complex turbulent flows combining with high-order numerical scheme of the Weighted Compact Nonlinear Scheme (WCNS) in the present study. It enables a seamless evolution from unsteady RANS to LES and finally approaches Direct Numerical Simulation (DNS) depending on the turbulent scales. In the framework of SATES, a new SATES-σ model with an adaptive model coefficient is developed by extending the underlying LES mode based on an enhanced sub-grid-scale model of the σ-model. The new SATES-σ is first examined in two benchmark cases of channel flow and flow past a square cylinder. Then, it is validated in supercritical flow past a circular cylinder to assess the performance of turbulent models. The results show significant improvements over the previous SATES and IDDES in the predictions of boundary layer flow. Finally, successful application is achieved in the accurate prediction of the stall of the MD-30P30N airfoil at a Reynolds number of 9×106 with wide angles of attack. The simulation results show good agreement with experimental results for surface pressure even for the challenging cases of 21 and 23 deg angles of attack. Again, the SATES-σ shows better results than the previous SATES and IDDES. The presented method has considerable potential for the challenging stall predictions. [ABSTRACT FROM AUTHOR]

In this paper, we present the development of a third-order density-based solver within the Open- FOAM framework, tailored for handling compressible flows. The solver incorporates implicit variational reconstruction on three-dimensional unstructured meshes, as well as a novel technology coupling reconstruction and time integration for both steady and unsteady simulations. To address the challenge of achieving high-order accuracy for curved geometries, we introduce a new approach for curved wall boundary reconstruction, specifically designed for situations where high-order mesh information is not readily available in OpenFOAM. Furthermore, we propose a simple WBAP limiter capable of capturing shocks without necessitating the whole-domain successive limiting procedure. Numerical tests were conducted to assess the solver's performance. The results reveal that our established solver exhibits higher accuracy in smooth flow simulations, while maintaining an excellent balance between accuracy and robustness for problems involving strong shocks and other complex flow structures. [ABSTRACT FROM AUTHOR]

We present the approaches to implementing the k − k L turbulence model within the framework of the high-order discontinuous Galerkin (DG) method. We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations. In order to enhance the robustness of approaches, some effective techniques are designed. The HWENO (Hermite weighted essentially non-oscillatory) limiting strategy is adopted for stabilizing the turbulence model variable k. Modifications have been made to the model equation itself by using the auxiliary variable that is always positive. The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods. Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations. [ABSTRACT FROM AUTHOR]

In this article, we present a finite element method for studying the dynamic behavior of deformable vesicles, which mimic red blood cells, in a non-Newtonian Casson fluid. The fluid membrane, represented by an implicit level-set function, adheres to the Canham–Helfrich model and maintains surface inextensibility constraint through penalty. We propose a two-step time integration scheme that incorporates higher-order accuracy by using an asymmetric composition of discrete flow based on the second-order backward difference formula, followed by a projection onto the real axis. Our framework incorporates variable time steps generated by an appropriate adaptation criterion. We validate our model through numerical simulations against existing experimental and numerical results in the case of purely Newtonian flow. Furthermore, we provide preliminary results demonstrating the influence of the non-Newtonian fluid model on membrane regimes. [ABSTRACT FROM AUTHOR]

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model, which cannot describe certain nonequilibrium phenomena by finite collisions of particles, decreasing the fidelity of the solution. Based on an alternative formulation of the targeted essentially non-oscillatory (TENO) scheme, a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems. The numerical flux is constructed by the TENO interpolation of the solution and its derivatives, instead of being reconstructed from the physical flux. The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations. The numerical methods are verified by several classical plasma problems. The results show that compared with the original TENO scheme, the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation. [ABSTRACT FROM AUTHOR]

The TENO-family schemes (Fu et al. in J Comput Phys 305: 333–359, 2016) have been demonstrated to perform well for compressible gas dynamics and turbulent flow predictions on structured meshes. However, the extension of the TENO schemes to unstructured meshes is non-trivial and challenging, particularly when the multiple design objectives are pursued simultaneously, i.e., restoring the high-order accuracy in smooth regions, retaining the low numerical dissipation for small-scale features, maintaining the sharp shock-capturing property, and featuring the good numerical robustness for high-Mach flows. In this work, a family of very-high-order (up to seventh-order accuracy) robust finite-volume TENO schemes with dual ENO-like stencil selection for unstructured meshes is proposed. The stencils include one large stencil and several small stencils. The novelty originates from a so-called dual ENO-like stencil selection strategy. Following a strong scale separation, the ENO-like stencil selection procedure with a small C T is first enforced among all the candidates such that the high-order candidate scheme on the large stencil is adopted for the final reconstruction when the local flow is smooth. If the large stencil is judged to be crossed by discontinuities, a second ENO-like stencil selection with a relatively large C T is applied to all the left small stencils and the ENO property is obtained by selecting the smooth small stencils which are not crossed by discontinuities. The smaller C T in the first stage ensures that the high-order reconstruction is restored for smooth flow scales with higher wavenumbers. On the other hand, the larger C T in the second stage can enforce a strong nonlinear adaptation for capturing discontinuities with better robustness. Such a dual ENO-like stencil selection strategy introduces an explicit scale separation and deploys the optimal strategy for different types of flows correspondingly. Without parameter tuning, a set of benchmark simulations has been conducted to validate the performance of the proposed TENO schemes. Numerical results demonstrate the good numerical robustness and the low-dissipation property for highly compressible flows with shockwaves. [ABSTRACT FROM AUTHOR]

A hybrid scheme is developed for direct numerical simulations of hypersonic flows over a blunt body. The scheme switches to the first-order AUSMPW+ scheme near the bow shock to provide sufficient dissipation to handle the carbuncle phenomenon. In the smooth part of the computational domain, a sixth-order central scheme with an eighth-order low-pass filter is adopted to provide high spatial accuracy to resolve turbulence. The hybrid scheme is shown to be able to obtain smooth and accurate predictions of laminar hypersonic flows over a blunt body. Using the hybrid scheme, a direct numerical simulation of a Mach 6 hypersonic flow over a circular cylinder is conducted. The result shows the turbulent structures in the near-wall region are well resolved by the hybrid scheme, and the bow shock is also captured without introducing any numerical oscillations. With the boundary layer transition on the cylinder's surface, the simulation indicates that the heat flux peak shifts from the stagnation point to the transitional zone and its peak value is increased by 50 % . [ABSTRACT FROM AUTHOR]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [ABSTRACT FROM AUTHOR]

Adaptive mesh refinement (AMR) and wavelet-based multi-resolution technique, which refine the spatial resolution in regions of interest and coarsen the mesh in other regions, are widely used in scientific computing for higher computational efficiency. The key component of AMR and the multi-resolution method is the adaptation strategy including the regularity estimation. In this paper, a new adaptation strategy for AMR and the multi-resolution method is proposed. Different from AMR, which measures the function regularity based on an empirical gradient operator, and the multi-resolution method with complicated wavelet analysis, the new approach examines the solution smoothness based on the high-order TENO reconstruction (Fu et al. in J Comput Phys 305:333–359, 2016). The core idea is that the TENO scheme not only provides the reconstructed data at the cell interface for flux evaluation but also classifies the local flow scales as smooth or nonsmooth on the discretized mesh. Since the scale-separation procedure is achieved in the spectral and characteristic space, the new adaptation strategy is weakly problem-dependent. Moreover, the extra complexity due to the empirical gradient computing or the wavelet analysis is eliminated. Based on Harten’s finite-volume multi-resolution method (Harten in J Comput Phys 115(2):319–338, 1994), a set of benchmark cases is simulated to validate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Over the past years, there has been innovative ideas about data-driven turbulence modeling proposed by scholars all over the world. This paper is a continuity of these significant efforts, with the aim of offering a better representation for turbulence physics. Previous works mainly focus on turbulence viscosity or Reynolds stress, while there are few works for turbulence transition. In our work, two mapping functions between average flow parameters and transition intermittency, a virtual physical quantity describing the amount of turbulence at a given position, are refactored, respectively, with neuron networks and random forests. These two functions are then coupled with the Spalart–Allmaras (SA) model to reconstitute two models for transition prediction. To demonstrate that these two coupled models provide improved prediction accuracy on transition compared with previous SA models, we conduct test cases all under a high-order weighted compact nonlinear scheme (WCNS). The prediction results of both coupled models significantly improved the capture of natural transitions occurring in the flows. Furthermore, the interpolation generalisation and extrapolation generalisation abilities of the coupled models are also demonstrated in this paper. The results emphasize the potential for machine learning as a supplementary in turbulence transition modeling. [ABSTRACT FROM AUTHOR]

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor's theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present an optimal fourth-order parameter-uniform non-monotone scheme on equidistributed meshes for singularly perturbed reaction–diffusion boundary value problems exhibiting boundary layers at both ends of the domain. We discretize the problem using a high-order non-monotone finite difference scheme and prove that the scheme is stable in the maximum norm. The equidistribution of an appropriate monitor function is used to generate the layer-adapted meshes to discretize the problem. The method is proved to be optimal fourth-order uniformly convergent on these equidistributed meshes. Numerical results are presented to validate the theory and to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305(2016): 333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, the candidate stencils include one large stencil and several small third-order stencils. Following a strong scale-separation procedure, a tailored novel ENO-like stencil selection strategy is proposed such that the high-order accuracy is restored in smooth regions by selecting the candidate reconstruction on the large stencil while the ENO property is enforced near discontinuities by adopting the candidate reconstruction from smooth small stencils. The nonsmooth stencils containing genuine discontinuities are explicitly excluded from the final reconstruction, leading to excellent numerical stability. Different from the WENO concept, such unique sharp stencil selection retains the low numerical dissipation without sacrificing the shock-capturing capability. The newly proposed framework enables arbitrarily high-order TENO reconstructions on unstructured meshes. For conceptual verification, the TENO schemes with third- to sixth-order accuracy are constructed. Without parameter tuning case by case, the performance of the proposed TENO schemes is demonstrated by examining a set of benchmark cases with broadband flow length scales. [ABSTRACT FROM AUTHOR]

In this article, we present a finite element method for studying the dynamic behavior of deformable vesicles, which mimic red blood cells, in a non-Newtonian Casson fluid. The fluid membrane, represented by an implicit level-set function, adheres to the Canham–Helfrich model and maintains surface inextensibility constraint through penalty. We propose a two-step time integration scheme that incorporates higher-order accuracy by using an asymmetric composition of discrete flow based on the second-order backward difference formula, followed by a projection onto the real axis. Our framework incorporates variable time steps generated by an appropriate adaptation criterion. We validate our model through numerical simulations against existing experimental and numerical results in the case of purely Newtonian flow. Furthermore, we provide preliminary results demonstrating the influence of the non-Newtonian fluid model on membrane regimes.

• A coupled high-order monotonicity-preserving interface capturing scheme for computations of compressible three-phase flows with phase changes is presented. • Numerical results are validated against experimental data. • The characteristics of bubble interactions with a free surface and parallel walls are discussed. • The effects of the initial pressure and non-condensable gas volume fraction on the behavior of the explosion bubble are analyzed. The complex dynamics of two flow problems of vapor explosion bubbles near a free surface and between two parallel plates are simulated by a compressible three-phase flow code. Firstly, a coupled high-order monotonicity-preserving (MP) interface capturing scheme is developed for the numerical computation of compressible three-phase flows, which is followed by validation with previous numerical and experimental data. Phase changes are taken into consideration and the empirical condensation coefficient is calibrated with the experimental data. For both two flow problems, good agreement between the numerical results and reference data is obtained. Most physical phenomena, including the free surface spike, the liquid jet before the impact and the complex interface deformations during multiple expanding and collapsing oscillations, are reasonably well reproduced. Afterwards, the effects of the initial vapor pressure and the non-condensable gas volume fraction in the surrounding water on the behavior of vapor bubbles are analyzed and discussed. It is found that both the initial vapor pressure and the non-condensable gas volume fraction have a substantial influence on the behavior of the bubble. Higher initial vapor pressure does indeed tend to result in larger bubbles. The presence of non-condensable gases tends to hinder the transfer of heat, slowing down condensation and potentially resulting in larger bubbles. When the initial vapor pressure is in the range of 10 to 20 atmospheres and the gas volume fraction is in the range of 0 to 0.2, the maximum vapor bubble volume is found to increase linearly with an increase of the amount of non-condensable gas. [ABSTRACT FROM AUTHOR]

Fu et al. (J Comput Phys 374:724–751, 2018), a class of adaptive targeted ENO (TENO) schemes have been proposed. Excellent resolution and robustness of the TENO scheme were validated by benchmark problems, but the computational cost is high. In order to improve the computational efficiency, a modified TENO scheme is proposed in this article. First, using the weighting strategy of the fifth-order TENO as an explicit discontinuity detector, locations of discontinuities are detected on the five-point stencil, and the initial target stencil with maximum support on the five-point stencil is obtained. Then, with a simple discontinuity detection strategy, the initial target stencil is enlarged point by point in the direction where no discontinuity is detected. In this manner, final target stencil with maximum support, not crossed by discontinuities, on the full six- or eight-point stencil is formed. Unlike TENO using candidate stencils with incrementally increased width, all stencils are three-point in the new framework. Due to the simplified determination of final target stencil and avoiding expensive calculation of smoothness indicator of large stencil, the computational cost of TENO-M is significantly lower than that of TENO. Several benchmark problems are conducted to validate the performance of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, an efficient low-dissipation high-order TENO scheme is proposed for ideal MHD flows. For high computational efficiency, a troubled-cell indicator based on the ENO-like stencil selection strategy in TENO schemes is introduced to isolate the discontinuities from smooth regions. While the high-order linear scheme is adopted for the smooth regions, a low-dissipation TENO scheme is applied for capturing discontinuities detected by the troubled-cell indicator. The case-independent parameters are given based on spectral analysis. Both the governing equations of the ideal MHD and the Hamilton–Jacobi type constrained transport equation for divergence-free condition can be solved by the newly proposed scheme. Since most computational regions are resolved by the linear scheme without expensive characteristic decomposition, flux splitting and nonlinear weight calculation, the proposed scheme is highly efficient. A set of benchmark cases has been simulated to demonstrate the performance of the proposed scheme. Numerical results reveal that remarkable speedup is achieved by the present scheme while the oscillation-free property and the high-order accuracy are preserved. [ABSTRACT FROM AUTHOR]

Over the past years, there has been innovative ideas about data-driven turbulence modeling proposed by scholars all over the world. This paper is a continuity of these significant efforts, with the aim of offering a better representation for turbulence physics. Previous works mainly focus on turbulence viscosity or Reynolds stress, while there are few works for turbulence transition. In our work, two mapping functions between average flow parameters and transition intermittency, a virtual physical quantity describing the amount of turbulence at a given position, are refactored, respectively, with neuron networks and random forests. These two functions are then coupled with the Spalart–Allmaras (SA) model to reconstitute two models for transition prediction. To demonstrate that these two coupled models provide improved prediction accuracy on transition compared with previous SA models, we conduct test cases all under a high-order weighted compact nonlinear scheme (WCNS). The prediction results of both coupled models significantly improved the capture of natural transitions occurring in the flows. Furthermore, the interpolation generalisation and extrapolation generalisation abilities of the coupled models are also demonstrated in this paper. The results emphasize the potential for machine learning as a supplementary in turbulence transition modeling.

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

Accurate prediction of real gas flows with strongly variable properties is an essential prerequisite for turbomachinery performance analysis. This paper presents a density-based solver for real gas flows in turbomachinery. The solver employs a third-order discretization scheme to improve computational accuracy. Real gas equations of state and look-up table are utilized in the solver to account for the strong nonlinear variations of thermophysical properties. The preconditioning method is adopted for simulations, including low-Mach flow regions, to address convergence difficulties. The implicit Time Consistent Preconditioned Gauss-Seidel Scheme (TCPGS) is employed for the time-stepping. The solver is validated by transonic compressor cascade wind tunnel experiments. Numerical test cases are adopted to evaluate the solver performance, including NASA Rotor 67, NASA Rotor 37, real gas flows in the Laval nozzle, and compressor seal. It has been proved that the numerical test results are consistent with published technical data. The advantages of this solver include numerical stability, computational efficiency, and physical accuracy, which indicate its applicability in the turbomachinery design process for real gases. • A high-order density-based solver for turbomachinery of the real gases is presented. • Strong changes in thermophysical properties of real gases can be calculated. • Preconditioning method is utilized for solving low-Mach flow regions. • The solver validation is conducted by transonic cascade wind tunnel experiments. • The performance of solver is evaluated by numerical tests. [ABSTRACT FROM AUTHOR]

The aim of this paper is the presentation of a new modified weighted essentially non-oscillatory scheme with a seventh-order of accuracy, and, a novel nonlinear weighting method. Local smoothness indicators of this scheme constructed using Lagrange's interpolation polynomial. We developed a new high-order global smoothness indicator, which guarantees the optimal order of accuracy of the scheme at critical points. We used the linear advection equation for verification of the order of accuracy at critical points. Later, we used a series of one- and two-dimensional benchmark problems to verify the shock-capturing capabilities and improved numerical resolution of the new scheme. [ABSTRACT FROM AUTHOR]

We propose high-order well-balanced finite-volume schemes for a broad class of hydrodynamic systems with attractive-repulsive interaction for ces and linear and nonlinear damping. Our schemes are suitable for free energies containing convolutions of an interaction potential with the density, which are essential for applications such as the Keller--Segel model, more general Euler--Poisson systems, or dynamic-density functional theory. Our schemes are also equipped with a nonnegative-density reconstruction which allows for vacuum regions during the simulation. We provide several prototypical examples from relevant applications highlighting the benefit of our algorithms and also elucidate some of our analytical results. [ABSTRACT FROM AUTHOR]

A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy than the Godunov-type scheme and can work well even in the resonant regime. Wave interaction problems are also tested where the scheme possesses a good accuracy. It turns out that the superbee limiter can provide us with more accurate approximations than van Leer’s limiter. [ABSTRACT FROM AUTHOR]

In this article, a new weight tri-diagonal iterative method in solving system of linear equation is proposed and its convergence is discussed. The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented. The numerical experiments shows that the proposed method demonstrates a better performance compared with weight Jacobi, successive-over relaxation and alternating group explicit methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, firstly, we solve the linear 3D Schrödinger equation using Douglas–Gunn alternating direction implicit (ADI) scheme and high-order compact (HOC) ADI scheme, which have the order O(τ2+h2) $O(\tau^{2}+h^{2})$ and O(τ2+h4) $O(\tau^{2}+h^{4})$, respectively. Secondly, a fourth-order compact ADI scheme, based on the Douglas–Gunn ADI scheme combined with second-order Strang splitting technique, is proposed for solving 3D nonlinear Schrödinger equation. Stability analysis has demonstrated that these schemes are unconditionally stable. Finally, numerical results show that these schemes preserve the conservation laws and provide accurate and stable solutions for the 3D linear and nonlinear Schrödinger equations.

The objective of this paper is to check the ability of the high-order OSMP scheme [Daru, V., and C. Tenaud. 2004. "HighOrder One-Step Monotonicity-Preserving Schemes for Unsteady Compressible Flow Calculations." Journal of Computational Physics 193 (2): 563–594] to accurately compute turbulent compressible flows with a special focus on the effect of the MP constraints on solutions of wall bounded turbulent shocked flows. We first validate numerical approaches on two canonical test cases: the 3D Taylor-Green vortex and the steady 2D shock-wave laminar boundary layer interaction. The emblematic case where a shock wave interacts with a turbulent boundary layer is then simulated. We conclude that the OSMP scheme coupled with a centred 2nd-order approximation for the diffusive fluxes constitutes a reliable numerical tool for the DNS of compressible turbulent shocked flows, that is completely competitive compared to other approaches since the overall errors are lowered of about one order of magnitude for the same grid resolution. [ABSTRACT FROM AUTHOR]

The monotonicity-preserving (MP) scheme is an accurate shock-capturing scheme. However, its performance is still inefficient for resolving high-frequency waves. In this paper, to improve the resolution characteristics, an upwind compact interpolation is proposed as a substitute to the original one in the MP scheme, and the coefficients of that were analytically optimized to minimize the dispersion and dissipation errors. Moreover, it was found that the limiting part of the original MP scheme degenerates the accuracy in a high-wavenumber region due to unnecessarily activation. This limitation is improved by applying a new indicator and criterion. The results of the nonlinear wave (N-wave) propagation demonstrate that the proposed scheme guarantees the robustness at the sharp discontinuity. At the same time, the solutions of linear wave propagation prove the excellent resolution of the proposed scheme. We intensively evaluated the performance for the standard and long-time situations of the shock-entropy wave interaction problems. The results prove that the usefulness of proposed scheme is more pronounced in the flow fields involving both of shock and waves. [ABSTRACT FROM AUTHOR]

A ( 2 N + 1 )-equation model to simulate the flow of N ( N ≥ 3 ) immiscible compressible fluids separated with interfaces is proposed. The model is based on the single velocity diffuse-interface method and includes N - 1 advection equations for fluid volume fractions. Solving the advection equations with a non-linear high-order scheme commonly results in the violation of the non-negativity constraint that any arbitrary partial sum of volume fractions should be in the interval [0, 1]. First, it is shown that this constraint can be met if the N - 1 advection equations are solved for some rational functions of volume fractions rather than for volume fractions themselves. The non-linear sub-cell slope reconstruction (MUSCL-type and THINC) with the proposed rational advection functions is proved to be non-oscillatory and provide the distribution of volume fractions satisfying the non-negativity constraint. Second, it is proved that the PV property (preservation of constant-pressure and constant-velocity equilibrium) is maintained providing that linear functions of volume fractions are used in the advection equations. We suggest two ways for resolving the contradiction in choosing the advection functions (functions of volume fractions) in accordance with the non-negativity constraint and the PV property. We also adopt two numerical methods—the Roe-type scheme and the HLLC scheme to solve the governing equations. Finally, the proposed numerical model is tested with several benchmark problems. The results obtained demonstrate robustness and effectiveness of the proposed numerical approach in solving multi-fluid flows with large interface deformations. [ABSTRACT FROM AUTHOR]

In this paper we present a modified seventh-order weighted essentially nonoscillatory scheme for hyperbolic conservation laws. Local smoothness indicators are constructed based upon Lagrange's interpolation polynomial. We constructed a new high-order global smoothness indicator to guarantee the scheme achieves optimal order of accuracy at critical points. We investigated this scheme at critical points and verified its order of convergence with the help of linear scalar test cases. We implemented it to various nonlinear scalar equations and system of Euler equations in one- and two-dimensions to demonstrate the discontinuity capturing and high resolution properties of the modified scheme. [ABSTRACT FROM AUTHOR]

The aim of this work is to construct efficient finite volume schemes for the numerical study of sediment transport in shallow water, in the framework of the Exner model [7,10]. In most cases, the velocity related to the sediment is much lower that the fluid velocity, which, in turn, may be much lower that the free-surface wave speed. Explicit methods that resolve all waves require small time steps due to the CFL stability restriction because of fast surface waves. Furthermore, if Rusanov flux is adopted, slow sediment waves may be affected by the large numerical diffusion. The objective of the present work is to drastically improve the efficiency in the computation of the evolution of the sediment by treating water waves implicitly, thus allowing much larger time steps than the one required by fully explicit schemes. The goal is reached by suitably semi-implicit schemes obtained by the use of implicit-explicit Runge-Kutta methods. [ABSTRACT FROM AUTHOR]

In this paper we blend the high order Compact Approximate Taylor (CAT) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 2D. The resulting scheme presents high accuracy on smooth solutions, essentially non-oscillatory behavior on irregular ones, and almost fail-safe property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behavior of the CAT-MOOD scheme. • Hig-order one-step Lax-Wendroff type scheme for 2D systems of conservation laws. • Positivity-preserving and conservative scheme. • Finite-difference scheme. • A-posteriori order adaptive scheme. • 2D numerical experiments as Euler equations with smooth and non-smooth initial conditions. [ABSTRACT FROM AUTHOR]

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. 03B80, 65L10, 65L12 [ABSTRACT FROM AUTHOR]

We present asymptotic methods for solving high frequency Helmholtz equations in anisotropic media. The methods are motivated by Babich's expansion that uses Hankel functions of the first kind to approximate the solution of high frequency Helmholtz equation in isotropic media. Within Babich's expansion, we can derive the anisotropic eikonal equation and a recurrent system of transport equations to determine the phase and amplitude terms of the wave, respectively. In order to reconstruct the wave with the phase and amplitude terms for any high frequencies, they must be computed with high-order accuracy, for which a high-order factorization approach based on power series expansions at the primary source is applied first to resolve the source singularities, after that high-order schemes can be implemented efficiently. Rigorous formulations are derived, and numerical examples are presented to demonstrate the methods. [ABSTRACT FROM AUTHOR]

To ensure time stability of a seventh-order dissipative compact finite difference scheme, fourth-order boundary closures are used near domain boundaries previously. However, this would reduce the global convergence rate to fifth-order only. In this paper, we elevate the boundary closures to sixth-order to achieve seventh-order global accuracy. To keep the improved scheme time stable, the simultaneous approximation terms (SATs) are used to impose boundary conditions weakly. Eigenvalue analysis shows that the improved scheme is time stable. Numerical experiments for linear advection equations and one-dimensional Euler equations are implemented to validate the new scheme. [ABSTRACT FROM AUTHOR]

Abstract We present a formulation of incompressible smoothed particle hydrodynamics (ISPH) method that utilizes pairwise-relaxing kernel to achieve approximately first-order consistency. Previous high-order formulations by using reproduced and corrected kernel function have had difficulties in ensuring momentum conservation. In the new scheme, relaxing constants for each kernel function are determined pair-wisely throughout the entire calculation domain by enforcing the Taylor-series consistency condition. We call this modified ISPH method Pairwise-Relaxing ISPH, or PR-ISPH. PR-ISPH retains high-order accuracy for non-uniform particle distributions. The spatial symmetry of the kernel function is kept in PR-ISPH thus momentum is strictly conserved. Several two-dimensional benchmark calculations are conducted to demonstrate the accuracy as well as the conservation property of the PR-ISPH. [ABSTRACT FROM AUTHOR]

Abstract We carried out a numerical study of a sonic transverse jet into a Ma = 2.0 crossflow based on a hybrid Reynolds-averaged Navier-Stokes (RANS)/large eddy simulation (LES) method. In order to enhance the calculation accuracy, an improved high-order finite-difference scheme is introduced. This scheme consists of an eighth-order central scheme and a nonlinear dissipation filter. The filter is derived from the explicit seventh-order weighted compact nonlinear scheme (WCNS-E−7). By incorporating a local smoothness indicator and a discontinuity sensor into the nonlinear filter, we further decrease the unnecessary dissipation and increase the resolution. With the proposed scheme, the key flow features, including shock structures, large-scale vortical structures and jet-crossflow shear/mixing layers, are well captured. The simulation shows a good agreement with the experiment data in jet penetration and diffusion. Highlights • RANS/LES simulation of a transverse jet in supersonic crossflow is performed. • An improved high-order nonlinear filter scheme is introduced. • Simulation reproduces the flowfield in good agreement with experiment. [ABSTRACT FROM AUTHOR]

A combination of two approaches to numerically solving second-order ODEs with a small parameter and singularities, such as interior and boundary layers, is considered, namely, compact high-order approximation schemes and explicit generation of layer resolving grids. The generation of layer resolving grids, which is based on estimates of solution derivatives and formulations of coordinate transformations eliminating the solution singularities, is a generalization of a method for a first-order scheme developed earlier. This paper presents formulas of the coordinate transformations and numerical experiments for first-, second-, and third-order schemes on uniform and layer resolving grids for equations with boundary, interior, exponential, and power layers of various scales. Numerical experiments confirm the uniform convergence of the numerical solutions performed with the compact high-order schemes on the layer resolving grids. By using transfinite interpolation or numerical solutions to the Beltrami and diffusion equations in a control metric based on coordinate transformations eliminating the solution singularities, this technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) electromagnetic (EM) modeling is an important tool for geophysical applications. In EM induction methods the modified Helmholtz equation is often used to describe scattered or residual electric fields in three dimensions. Throughout this paper, a high order compact finite difference scheme for the solution of that equation for vertical magnetic dipole source (VMD) is presented. The approximation of the residual electric field intensity using a fourth order compact finite difference (FD) discretizer is achieved with the solution of a block linear system, the coefficient matrices are large and sparse with a particular structure, implying the application of a matrix-free Krylov subspace method for an efficient numerical solution. The proposed solver is being examined using a number of test problems in uniform and non-uniform grid spacing where numerical and analytical solutions for the homogeneous half-space cases are being compared. • A parallel algorithm for the solution of the modified Helmholtz equation used to describe scattered 3D electric fields for a VMD source • A 4th order compact finite difference numerical scheme is being used • The method presented proves to be efficient without the application of PMLs, preconditioning or divergence correction techniques • The implementation solves scattered fields using a uniform or a non-uniform grid and small-sized cells [ABSTRACT FROM AUTHOR]

Firstly, an improved TENO scheme based on perturbed polynomial reconstruction (TENO-P) is proposed for hyperbolic conservation laws on the foundation of fifth-order TENO5 scheme. Two different free-parameters are introduced into the reconstruction coefficients to eliminate the third-order errors in the candidate stencils and the fifth-order error in the global stencil, respectively. One-order of accuracy improvement is achieved by adaptively adjusting the values of the free-parameters, resulting a maximum sixth-order accuracy in the smooth stencil. Secondly, to improve the accuracy order of the fifth-order TENO5-THINC scheme, a sixth-order TENO-THINC-P scheme is proposed by combining the proposed TENO-P scheme for smooth regions with the THINC reconstruction for non-smooth discontinuities. Compared to the TENO5 scheme and the TENO5-THINC scheme, the numerical results show the dissipations of the TENO-P scheme and the TENO-THINC-P scheme are reduced, and their resolutions are significantly improved in resolving complicated flow structures with multiple discontinuities. Taking the computational efficiency and the numerical resolution into consideration, the TENO-P scheme is suitable for resolving vortex structures of turbulence while the TENO-THINC-P scheme is more suitable for resolving vortex structures near strong discontinuities. The derivation processes of the TENO-P scheme and the TENO-THINC-P scheme also demonstrate that the perturbed polynomial reconstruction has simple principle and good portability and expandability, so the idea of the TENO-P scheme and the TENO-THINC-P scheme can be easily extended to other improved very-high-order TENO schemes or WENO schemes. • TENO schemes (TENO-P, TENO-THINC-P) with improved accuracy order are proposed by using perturbed polynomial reconstruction. • A maximum sixth-order accuracy method is obtained in smooth stencil by introducing a free-parameter in six-point stencil. • The TENO-P scheme is suitable for resolving vortex structures of turbulence. • The TENO-THINC-P scheme is more suitable for resolving vortex structures near strong discontinuities. • The perturbed polynomial reconstruction method has simple principle and good portability and expandability. [ABSTRACT FROM AUTHOR]