Flux Splitting Method Research Articles

A novel flux splitting based on wave-particle splitting for ideal magnetohydrodynamics

A novel flux splitting method based on wave-particle splitting is developed for one-dimensional ideal magnetohydrodynamics. While ideal magnetohydrodynamics (MHD) equations are non-convex with non-homogeneous flux as opposed to their hydrodynamic counterparts, the present flux splitting methods cannot develop Riemann solver. The proposed approach based on wave-particle method referred as Advection Magnetic-direction Wave Particle Splitting (AMWPS) scheme splits the flux into four parts: the wave-like total pressure part, the wave-like normal magnetic part, the particle-like advection part and the particle-like tangential magnetic part. The particle-like parts can be solved by the scheme developed by Toro and Vazquez-Cendon (TV scheme) and we propose a novel approach to the solution of Riemann problem formed by the wave-like parts including total pressure sub-flux and normal magnetic sub-flux. As for the ordered wave foliation in MHD and other limitations in wave structure, we also analyze the relation between plasma β and specific heat ratio γ. The advantage of AMWPS is that it can sharply capture isolated, stationary Alfven wave discontinuity. Several one-dimensional and two-dimensional MHD problems have been tested to highlight the accuracy, positivity preservation and robustness of AMWPS scheme and comparative studies show that AMWPS significantly outperforms the Riemann solver named Harten, Lax and Leer for contact wave (HLLC) in most cases.

Recasting an operator splitting solver into a standard finite volume flux-based algorithm. The case of a Lagrange-projection-type method for gas dynamics

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes.

A New Flux Splitting Method and Central Difference Scheme for Magnetohydrodynamics

In this work, we describe a new flux splitting method for an ideal MHD in the 3D curvilinear coordinate system. The method is designed to create a dissipation term by zero flux term of flux split and create a new central difference scheme for ideal MHD. Numerical tests are performed to demonstrate the accuracy and capability of the new scheme to capture complex waves of ideal MHD.

Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model

The water hammer phenomenon is the main problem in long-distance pipeline networks. The MOC (Method of characteristics) and finite difference methods lead to severe constraints on the mesh and Courant number, while the finite volume method of the second-order Godunov scheme has limited intermittent capture capability. These methods will produce severe numerical dissipation, affecting the computational efficiency at low Courant numbers. Based on the lax-Friedrichs flux splitting method, combined with the upstream and downstream virtual grid boundary conditions, this paper uses the high-precision fifth-order WENO scheme to reconstruct the interface flux and establishes a finite volume numerical model for solving the transient flow in the pipeline. The model adopts the GPU parallel acceleration technology to improve the program’s computational efficiency. The results show that the model maintains the excellent performance of intermittent excitation capture without spurious oscillations even at a low Courant number. Simultaneously, the model has a high degree of flexibility in meshing due to the high insensitivity to the Courant number. The number of grids in the model can be significantly reduced and higher computational efficiency can be obtained compared with MOC and the second-order Godunov scheme. Furthermore, this paper analyzes the acceleration effect in different grids. Accordingly, the acceleration effect of the GPU technique increases significantly with the increase in the number of computational grids. This model can support efficient and accurate fast simulation and prediction of non-constant transient processes in long-distance water pipeline systems.

An efficient discrete velocity method with inner iteration for steady flows in all flow regimes

An efficient improved discrete velocity method (IDVM) with inner iteration is presented to simulate the steady flows in all flow regimes in this work. It is an extension of our previous implicit IDVM to achieve a faster convergence rate. In the previous method, both the discrete velocity Boltzmann equation (DVBE) and the corresponding macroscopic governing equations are solved synchronously, where the computational discrete cost is dominated by the calculation of the DVBE since the number of distribution functions is far larger than that of macroscopic conservative variables. Furthermore, the convergence rate of the calculation of the DVBE is affected by the predicted equilibrium state obtained from the solution of macroscopic governing equations. To provide a more accurate predicted equilibrium state for the fully implicit discretization of the DVBE, an inner iteration is introduced into the solution of macroscopic governing equations, and the flux Jacobian of these equations is evaluated by the difference of numerical fluxes of Navier–Stokes equations rather than the Euler equation-based flux splitting method used in the previous implicit IDVM. This more accurate prediction procedure endows the developed method to accelerate the computation greatly, especially in the continuum flow regime. Numerical results indicate that, in the continuum flow regime, the present method is about one order of magnitude faster than the previous implicit IDVM and one to two orders of magnitude faster than the conventional semi-implicit DVM.

On Improving Algorithm Efficiency of Gas-Kick Simulations toward Automated Influx Management: A Robertson Differential-Algebraic-Equation Problem Approach

Summary Improved numerical efficiency in simulating wellbore gas-influx behaviors is essential for realizing real-time model-prediction-based gas-influx management in wells equipped with managed-pressure-drilling (MPD) systems. Currently, most solution algorithms for high-fidelity multiphase-flow models are highly time consuming and are not suitable for real-time decision making and control. In the application of model-predictive controllers (MPCs), long calculation time can lead to large overshoots and low control efficiency. This paper presents a drift-flux-model (DFM)-based gas-influx simulator with a novel numerical scheme for improved computational efficiency. The solution algorithm to a Robertson problem as differential algebraic equations (DAEs) was used as the numerical scheme to solve the control equations of the DFM in this study. The numerical stability and computational efficiency of this numerical scheme and the widely used flux-splitting methods are compared and analyzed. Results show that the Robertson DAE problem approach significantly reduces the total number of arithmetic operations and the computational time compared with the hybrid advection-upstream-splitting method (AUSMV) while maintaining the same prediction accuracy. According to the “Big-O notation” analysis, the Robertson DAE approach shows a lower-order growth of computational complexity, proving its good potential in enhancing numerical efficiency, especially when handling simulations with larger scales. The validation of both the numerical schemes for the solution of the DFM was performed using measured data from a test well drilled with water-based mud (WBM). This study offers a novel numerical solution to the DFM that can significantly reduce the computational time required for gas-kick simulation while maintaining high prediction accuracy. This approach enables the application of high-fidelity two-phase-flow models in model-prediction-based decision making and automated influx management with MPD systems.

A new diffusion-regulated flux splitting method for compressible flows

An improved version of a Low-Diffusion-Flux-Splitting (LDFS) scheme is presented. The original scheme was robust and accurate, especially in capturing strong shocks. However, the present studies reveal that a scope exists to improve the accuracy of the scheme to capture weak shocks. In order to meet this objective, the new methodology introduced in the present work imparts the LDFS scheme with the capability to crisply capture shocks and contact discontinuities irrespective of their strengths or grid inclinations. This multidimensional-like effect is imparted to the original scheme by first converting its upwind formulation to a central scheme with an artificial-viscosity term, followed by regulating the numerical diffusion using a Diffusion Regulation (DR) parameter and a shock switch. To the best of the authors’ knowledge this is the first successful attempt to apply this DR model with a shock switch to a Flux-Vector-Splitting scheme. The switch acts as the parameter for deciding where to locally activate or deactivate the diffusion regulation. The present innovation is found to significantly improve the shock-capturing accuracies of the parent scheme, especially for grid inclined weak shocks. Simultaneously it ensures that the new scheme is as robust as the parent scheme, especially in resolving strong shocks. The merits of the new methodology are clearly shown with the help of a number of carefully chosen inviscid and viscous test cases at a wide range of Mach numbers.

Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension

We introduce and analyse the so-calledReference Solution IMplicit-EXplicit schemeas a flux-splitting method for singularly-perturbed systems of balance laws. RS-IMEX scheme’s bottom-line is to use the Taylor expansion of the flux function and the source term around a reference solution (typically the asymptotic limit or an equilibrium solution) to decompose the flux and the source into stiff and non-stiff parts so that the resulting IMEX scheme is Asymptotic Preserving (AP) w.r.t. the singular parameter tending to zero. We prove the asymptotic consistency, asymptotic stability, solvability and well-balancing of the scheme for the case of the one-dimensional shallow water equations when the singular parameter is the Froude number. We will also study several test cases to illustrate the quality of the computed solutions and to confirm the analysis.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

一种健壮的低耗散通量分裂格式

With the rapid development of computational fluid dynamics, it is particularly important to design accurate, efficient and robust numerical schemes. Through the characteristics analyses of 3 popular flux splitting methods (AUSM, Zha-Bilgen and Toro-Vázquez), a simple, low-dissipation and robust flux splitting scheme (named as R-ZB) was constructed. The flux of Euler equations was split into a convection flux and a pressure flux with the Zha-Bilgen splitting procedure. The convection flux was computed with a simple upwinding scheme, and the pressure flux was evaluated with a low-dissipation HLL scheme to overcome the flaw of failing to capture contact discontinuities. Numerical experiments show that, the proposed R-ZB scheme not only retains the merits of the original Zha-Bilgen scheme, such as simpleness, efficiency and capturing contact discontinuities accurately, etc., but also has better robustness, which eliminates the numerical shock instabilities in the calculation of 2D problems.

A hybrid flux splitting method for compressible flow

ABSTRACTA hybrid flux splitting scheme, called AUSM+–FVS((advection upstream splitting method)+–flux vector splitting), is proposed in this article to calculate the inviscid fluxes of Euler/Navier–Stokes equations. This new scheme is obtained by hybridizing the AUSM+ scheme with FVS method. When the local Mach number tends to zero, this scheme is similar to the AUSM+. Contrarily, this scheme is similar to the FVS at the shock region. Thus, this scheme has the accuracy of AUSM+ in boundary layer region and the robustness of FVS in shock region. Several numerical tests show that AUSM+–FVS can reduce the shock instability and has a good accuracy in boundary layer region.

An immersed boundary formulation for simulating high-speed compressible viscous flows with moving solids

We present a robust sharp-interface immersed boundary method for numerically studying high speed flows of compressible and viscous fluids interacting with arbitrarily shaped either stationary or moving rigid solids. The Navier–Stokes equations are discretized on a rectangular Cartesian grid based on a low-diffusion flux splitting method for inviscid fluxes and conservative high-order central-difference schemes for the viscous components. Discontinuities such as those introduced by shock waves and contact surfaces are captured by using a high-resolution weighted essentially non-oscillatory (WENO) scheme. Ghost cells in the vicinity of the fluid–solid interface are introduced to satisfy boundary conditions on the interface. Values of variables in the ghost cells are found by using a constrained moving least squares method (CMLS) that eliminates numerical instabilities encountered in the conventional MLS formulation. The solution of the fluid flow and the solid motion equations is advanced in time by using the third-order Runge–Kutta and the implicit Newmark integration schemes, respectively. The performance of the proposed method has been assessed by computing results for the following four problems: shock-boundary layer interaction, supersonic viscous flows past a rigid cylinder, moving piston in a shock tube and lifting off from a flat surface of circular, rectangular and elliptic cylinders triggered by shock waves, and comparing computed results with those available in the literature.

A robust flux splitting method with low dissipation for all‐speed flows

SummaryIn this paper, we propose a simple robust flux splitting method for all‐speed flows with low dissipation. Following the Toro‐Vazquez splitting, the inviscid flux is split into the convective and the pressure parts first. Then, we apply the Harten‐Lax‐van Leer (HLL) algorithm to each parts with low dissipation modification. Here, the modification improves the method with accurately resolving contact discontinuity and high resolution for low speed flows. Several carefully chosen numerical tests are investigated, and the results show that the proposed scheme is capable of resolving contact discontinuity, robust against the shock anomaly and accurate at all‐speeds. Because of these properties, it is expected to be widely applied to all‐speed flow studies. Copyright © 2016 John Wiley & Sons, Ltd.

A flux splitting method for the Baer–Nunziato equations of compressible two-phase flow

Here we extend the Toro–Vázquez flux vector splitting approach (TV), originally proposed for the ideal 1D Euler equations in [1], to the Baer–Nunziato equations of compressible two-phase flow. Following the TV approach we identify corresponding advection and pressure operators. We perform a rigorous analysis of the associated non-conservative pressure system and derive its complete characteristic structure. The choice of the advection numerical flux is obvious. For the pressure system, several schemes are presented. The complete schemes are then implemented in the setting of finite volume and path-conservative methods and are systematically assessed in terms of accuracy and efficiency, through a carefully selected suite of test problems. The presented schemes constitute a building block for the construction of high-order numerical methods for solving the Baer–Nunziato equations. Here, as an illustrative example of such possibility, we present the construction of a second-order scheme.

A flux-splitting method for hyperbolic-equation system of magnetized electron fluids in quasi-neutral plasmas

A flux-splitting method is proposed for the hyperbolic-equation system (HES) of magnetized electron fluids in quasi-neutral plasmas. The numerical fluxes are split into four categories, which are computed by using an upwind method which incorporates a flux-vector splitting (FVS) and advection upstream splitting method (AUSM). The method is applied to a test calculation condition of uniformly distributed and angled magnetic lines of force. All of the pseudo-time advancement terms converge monotonically and the conservation laws are strictly satisfied in the steady state. The calculation results are compared with those computed by using the elliptic–parabolic-equation system (EPES) approach using a magnetic-field-aligned mesh (MFAM). Both qualitative and quantitative comparisons yield good agreements of results, indicating that the HES approach with the flux-splitting method attains a high computational accuracy.

A low diffusion flux splitting method for inviscid compressible flows

A low diffusion flux splitting method capable of capturing crisp shock profile and exact contact surface is presented. Here, the flux vector of the Euler equations is split into convective and pressure parts following Toro–Vázquez formulation. The low diffusive property of the present scheme is brought about by adding an anti-diffusion term to the pressure parts. This numerical method can be regarded as an improved version of Toro and Vázquez’s flux splitting schemes (i.e. TV and TV-AWS) which are found to produce shock instabilities and carbuncle phenomena. Numerical results for several carefully chosen one- and two-dimensional test problems are investigated to demonstrate the accuracy and robustness of the proposed scheme.

On the accuracy and robustness of a new flux splitting method

The high speed flow problems usually involve complex flow phenomena, such as strong shock waves, shock-shock interactions, and shear layers. Prediction of these problems requires robust, efficient and accurate numerical methods. A robust flux splitting method capable of capturing crisp shock profile and exact contact surface is presented. Here, the flux vector of the Euler equation is split into convective and pressure parts according to the Toro's formulation. The accuracy of the numerical method at the contact discontinuity is examined first. Sufficient conditions of shock stability for this new method are obtained through a linear perturbation analysis. Several carefully chosen test problems are numerically investigated, and the numerical results demonstrate the accuracy and robustness of the proposed scheme.

A flux splitting method for the Euler equations

The solution of the Euler and Navier–Stokes equations was the focus of this research. As with methods such as the AUSM-family, this work further attempted to construct an explicit upwind method that combines both simplicity and accuracy. In particular, with an appropriate choice of working variables, the aim was to develop a simple method that is based on the advection of the characteristic variables. The end result is a computationally efficient method valid for all homogeneous fluids with accuracy comparable to that of expensive flux-difference splitting methods. Good performance was extensively verified with Riemann test problems and results were on par with Roe's scheme. Furthermore, at least one formulation of the method passed all test problems satisfactory without requiring manual intervention. The method also illuminates some similarities between flux-difference splitting and the AUSM-family of methods.

The Analysis of a Transmission Line Crosstalk in the Time Domain Based on First-Order Upwind

This article will combine the difference scheme of first-order upwind with the multi-conductor transmission lines equation to analysis the multi-conductor transmission lines crosstalk in the time domain. First-order upwind is a finite difference algorithm in the time domain; it has a first order accuracy, in the discontinuous solution there is no non-physics-oscillation, when simulate the signal. The flux splitting method which is applied to the first-order upwind solved the problem that the characteristic line direction of the wind type make plus or minus transformation along with the coefficient, make the programming simple. In this paper, simulation results of transmission line crosstalk in this algorithm will be compared with the traditional leapfrog scheme, to verify its effectiveness.

Transient Analysis of Multi-Conductor Transmission Line Based on the Second-Order Upwind Scheme

On the basis of the second-order upwind scheme, combined with the flux splitting method, we derive a uniform multi-conductor transmission line time-domain calculations. This method is a direct time-domain discrete numerical method with second order accuracy. It does not have any special requirements for circuit and the application process and complex transformations which lead to facilitate the preparation of the program. Numerical experiments show that this method of preparation of general purpose computing program has roughly the same computational efficiency with the traditional leapfrog scheme and at the discontinuous solutions, there are no non-physical oscillations. It can easily be used to calculate a uniform multi-conductor transmission line coupling.