A Conservative Modification to the Ghost Fluid Method for Compressible Multiphase Flows

2011 ◽  
Vol 10 (4) ◽  
pp. 785-806 ◽  
Author(s):  
Wei Liu ◽  
Li Yuan ◽  
Chi-Wang Shu
Keyword(s):  
Multiphase Flows ◽  
Time Discretization ◽  
Weno Scheme ◽  
Third Order ◽  
Ghost Fluid Method ◽  
Material Interface ◽  
Numerical Resolution ◽  
Runge Kutta Method ◽  
Modification Procedure ◽  
Fifth Order

AbstractA conservative modification to the ghost fluid method (GFM) is developed for compressible multiphase flows. The motivation is to eliminate or reduce the conservation error of the GFM without affecting its performance. We track the conservative variables near the material interface and use this information to modify the numerical solution for an interfacing cell when the interface has passed the cell. The modification procedure can be used on the GFM with any base schemes. In this paper we use the fifth order finite difference WENO scheme for the spatial discretization and the third order TVD Runge-Kutta method for the time discretization. The level set method is used to capture the interface. Numerical experiments show that the method is at least mass and momentum conservative and is in general comparable in numerical resolution with the original GFM.

Numerical Simulations of a Polydisperse Sedimentation Model by Using Spectral WENO Method with Adaptive Multiresolution

2016 ◽  
Vol 13 (06) ◽  
pp. 1650037
Author(s):  
Carlos A. Vega ◽  
Francisco Arias
Keyword(s):  
Convergence Rates ◽  
Time Discretization ◽  
Strong Stability ◽  
Kutta Method ◽  
Strong Stability Preserving ◽  
Third Order ◽  
Runge Kutta Method ◽  
Sedimentation Model ◽  
Runge Kutta ◽  
Fifth Order

In this work, we apply adaptive multiresolution (Harten’s approach) characteristic-wise fifth-order Weighted Essentially Non-Oscillatory (WENO) for computing the numerical solution of a polydisperse sedimentation model, namely, the Höfler and Schwarzer model. In comparison to other related works, time discretization is carried out with the ten-stage fourth-order strong stability preserving Runge–Kutta method which is more efficient than the widely used optimal third-order TVD Runge–Kutta method. Numerical results with errors, convergence rates and CPU times are included for four and 11 species.

Simulation of Water Droplet Merging under Shock Wave Using Real Ghost Fluid Method

2013 ◽  
Vol 444-445 ◽  
pp. 628-632
Author(s):  
Ru Chao Shi ◽  
Sheng Li Xu ◽  
Ya Jun Zhang
Keyword(s):  
Shock Wave ◽  
Interpolation Method ◽  
Time Discretization ◽  
Problem Solver ◽  
Ghost Fluid Method ◽  
Numerical Accuracy ◽  
Air Water Interface ◽  
Fifth Order ◽  
The Given ◽  
Flow States

This paper presents a 3D numerical simulation of water droplets merging under a given shock wave. We couple interpolation method to RGFM (Real Ghost Fluid Method) to improve the numerical accuracy of RGFM. The flow states of air-water interface are calculated by ARPS (approximate Riemann problem solver). Flow field is solved by Euler equation with fifth-order WENO spatial discretization and fourth-order R-K (Runge-Kutta) time discretization. We also employ fifth-order HJ-WENO to discretize level set equation to keep track of gas-liquid interface. Numerical results demonstrate that droplets shape has little change before merging and the merged droplet gradually becomes umbrella-shaped under the given shock wave. We verify that combination of RGFM with interpolation method has the property of reducing numerical error by comparing to the results without employment of interpolation method.

High-order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates

2015 ◽  
Vol 25 (08) ◽  
pp. 1553-1588 ◽  
Author(s):  
Yan Jiang ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Random Walk ◽  
Finite Difference ◽  
Temporal Integration ◽  
Time Discretization ◽  
High Order ◽  
Correlated Random Walk ◽  
Third Order ◽  
Positivity Preserving ◽  
Fifth Order ◽  
High Order Finite Difference

In this paper, we discuss high-order finite difference weighted essentially non-oscillatory schemes, coupled with total variation diminishing (TVD) Runge–Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintenance of third-order spatial/temporal accuracy when the limiters are applied to a third-order finite difference scheme and third-order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth-order accuracy.

scholarly journals The accuracy of finite-difference schemes calculating the interaction of shock waves

2019 ◽  
Vol 485 (6) ◽  
pp. 691-696 ◽  
Author(s):  
V. V. Ostapenko ◽  
N. A. Khandeeva
Keyword(s):  
Finite Difference ◽  
Shock Waves ◽  
Difference Schemes ◽  
Weno Scheme ◽  
Shock Capturing ◽  
Finite Difference Schemes ◽  
Third Order ◽  
Interaction Of Shock Waves ◽  
Fifth Order

The accuracy with which the shock-capturing finite-difference schemes calculate the flows with interaction of shock waves is studied. It is shown that, in the domains between the shock waves after their incidence, the calculation accuracy of invariants of the combined schemes is several orders of magnitude higher than the accuracy of the WENO-scheme, which is fifth-order in space and third-order in time.

High Resolution Numerical Simulation of Detonation Diffraction of Condensed Explosives

2015 ◽  
Vol 12 (02) ◽  
pp. 1550005 ◽  
Author(s):  
Cheng Wang ◽  
Xinqiao Liu
Keyword(s):  
Numerical Simulation ◽  
High Resolution ◽  
Dead Zone ◽  
Third Order ◽  
Density Region ◽  
Specific Expression ◽  
Runge Kutta Method ◽  
Condensed Explosives ◽  
Pressure Region ◽  
Fifth Order

In this paper, the specific expression for pressure and sound speed in chemical reaction zone of condensed explosives are theoretically deduced, and a new method for deriving the partial derivative of pressure in respect of every conserved quantity is proposed. Combined with the third-order TVD Runge–Kutta method, we develop a parallel solver using the fifth-order high-resolution weighted essentially non-oscillatory (WENO) finite difference scheme to simulate detonation diffraction for two-dimensional condensed explosives. The numerical simulation results revealed the forming reasons of the low-pressure region, the low-density region, the "vortex" region and the "dead zone" in the vicinity of the corner. Furthermore, it demonstrated that the retonation will generate along the inner wall, and it plays an important role in the process of detonation diffraction.

High Resolution Numerical Simulation of Shock-to-Detonation Transition of Condensed-Phase Explosives

2013 ◽  
Vol 767 ◽  
pp. 40-45
Author(s):  
Wang Cheng ◽  
Xin Qiao Liu ◽  
Jian Guo Ning
Keyword(s):  
High Resolution ◽  
Condensed Phase ◽  
Pulse Width ◽  
Propagation Distance ◽  
Weno Scheme ◽  
Runge Kutta Method ◽  
Two Factors ◽  
Detonation Transition ◽  
Condensed Phase Explosives ◽  
Fifth Order

In this paper, shock-to-detonation transition for condensed phase explosives is numerically simulated by adopting high resolution numerical scheme. Fifth-order WENO scheme and third-order TVD Runge-Kutta method are employed to discretize Euler equations with chemical reaction source in space and time respectively, and parallel high resolution code is developed. Applying this code, the influence of incident pressure and pulse width on the run distance to detonation is investigated. The numerical results show that incident pressure and pulse width govern the initiation process. If appropriate values are taken for incident pressure and pulse width, the pressure will increase with the enlarging of the shock wave propagation distance, and finally the explosives reach steady detonation. The run distance to detonation is also influenced by those two factors, and it gets shorter with the increase of pulse width and incident pressure. When the incident pressure and the pulse width are small enough, the retonation phenomenon can be observed, and it becomes obvious along with the decreasing of incident pressure and pulse width.

scholarly journals High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Yanqun Jiang ◽  
Xun Chen ◽  
Xu Zhang ◽  
Tao Xiong ◽  
Shuguang Zhou
Keyword(s):  
Mach Number ◽  
Gas Dynamics ◽  
Compressible Flows ◽  
Time Discretization ◽  
Euler System ◽  
High Order ◽  
Shock Capturing ◽  
Third Order ◽  
High Mach ◽  
Fifth Order

AbstractThe computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

scholarly journals A New Smoothness Indicator of Adaptive Order Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 69
Author(s):  
Omer Musa ◽  
Guoping Huang ◽  
Mingsheng Wang
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Convex Combination ◽  
Computational Cost ◽  
Weno Scheme ◽  
Third Order ◽  
Simple Formulation ◽  
Order Linear ◽  
Linear Reconstruction ◽  
Smoothness Indicator ◽  
Fifth Order

Adaptive order weighted essentially non-oscillatory scheme (WENO-AO(5,3)) has increased the computational cost and complexity of the classic fifth-order WENO scheme by introducing a complicated smoothness indicator for fifth-order linear reconstruction. This smoothness indicator is based on convex combination of three third-order linear reconstructions and fifth-order linear reconstruction. Therefore, this paper proposes a new simple smoothness indicator for fifth-order linear reconstruction. The devised smoothness indicator linearly combines the existing smoothness indicators of third-order linear reconstructions, which reduces the complexity of that of WENO-AO(5,3) scheme. Then WENO-AO(5,3) scheme is modified to WENO-O scheme with new and simple formulation. Numerical experiments in 1-D and 2-D were run to demonstrate the accuracy and efficacy of the proposed scheme in which WENO-O scheme was compared with original WENO-AO(5,3) scheme along with WENO-AO-N, WENO-Z, and WENO-JS schemes. The results reveal that the proposed WENO-O scheme is not only comparable to the original scheme in terms of accuracy and efficacy but also decreases its computational cost and complexity.

scholarly journals A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
M. Mechee ◽  
N. Senu ◽  
F. Ismail ◽  
B. Nikouravan ◽  
Z. Siri
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Film Flow ◽  
New Method ◽  
Kutta Method ◽  
Thin Film Flow ◽  
Third Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

scholarly journals High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

2020 ◽  
Author(s):  
Yanqun Jiang ◽  
Xun Chen ◽  
Xu Zhang ◽  
Tao Xiong ◽  
Shuguang Zhou
Keyword(s):  
Mach Number ◽  
Gas Dynamics ◽  
Compressible Flows ◽  
Time Discretization ◽  
Euler System ◽  
High Order ◽  
Shock Capturing ◽  
Third Order ◽  
High Mach ◽  
Fifth Order

Abstract The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe CFL restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization and a fifth-order WCNS is used for the spatial discretization. The designed semi-implicit WCNS is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed method.

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