Third-order TVD Runge Kutta Method Research Articles

Modelling study of the spread of liquid fuel spill fires using shallow water equations

In oil spill–fire accidents, the spread of burning fuel from petroleum product containers may increase accident losses and trigger further accidents. Studying the spreading process of spill fires is important to prevent accidents and mitigate losses. Herein, a spill-fire spread model was proposed to predict the spread rate and time varying burning area of liquid fuel spill fires. The model was developed using one-dimensional nonlinear shallow water equations. Fuel consumption was used as the source term in the equations. To ensure numerical stability and accuracy, the third-order TVD Runge–Kutta method and the second-order MUSCL scheme were used for time advancement and numerical space reconstruction, respectively. Numerical flux was calculated by using the HLL approximate Riemann solver. The algorithm of the shallow water equations was modified in accordance with the spread characteristics of spill fires. Using the developed spread model, the dynamic burning area and dynamic fuel thickness of a spill fire were calculated. The simulation results were verified using the experimental results on continuously released n-heptane spill fires, and the two results were in good agreement.

Study on Burning Surface Regression Algorithm under Erosive Burning Based on CT Images of Solid Rocket Motor Grain

The presence of the erosive burning effect during the operation of a solid rocket motor (SRM) is one of the most important factors affecting the proper operation of the motor. To solve the effects of the operating process of the motor under erosive burning, a synthesis algorithm based on the actual CT images is proposed to combine the Level-set (LS) method with the minimum distance function (MDF) method for the simulation of the burning surface regression of the grain under erosive burning. The Hamilton–Jacobi control equation can be solved exactly for the discrete form of LS. To improve the computational efficiency of the LS method, the minimum distance field is initialized and only the distortion grid is adjusted during the reinitialization. The third-order TVD Runge–Kutta method is used to solve the problem of numerical oscillation and improve the calculation accuracy. The experiments simulate the burning process of the NAWC No. 6 partial grain under erosive burning, which can provide the main basis for the performance design of solid propellant. The experimental results show that the method has good applicability to three-dimensional complex grains. It can realize the simulation of grains under erosive burning and its calculation accuracy is high.

Direct WENO scheme for dispersion-type equations

In this paper, we present a weighted essentially non-oscillatory (WENO) scheme for dispersive equations which may generate physical high-frequency oscillation in the non-smooth interface. The third derivative term is approximated directly by a conservative flux difference. A finite-difference WENO scheme of fifth-order is constructed for the discretization of spatial differentiation. The wave behavior of linear and nonlinear dispersion equations is simulated by using the proposed scheme in space direction and the third-order TVD Runge–Kutta method in the time direction. Numerical examples demonstrate the accuracy and good performance of the proposed scheme.

A hybrid kinetic WGVC–WENO scheme for compressible flows

In this paper, we propose a hybrid kinetic weighted group velocity control–weighted essentially non-oscillatory (WGVC–WENO) scheme which combines the WGVC–WENO reconstruction technique He et al. (2014) with the hybrid kinetic method proposed in Liu et al. (2015). The new scheme is based on the idea of hybridization. The hybrid kinetic method is adopted for flux calculations, and the WGVC–WENO technique is adopted for physical reconstructions. In smooth flow regions, the collision-related flux method and WGVC reconstruction technique play a leading role. The kinetic flux vector splitting (KFVS) method and WENO reconstruction technique are dominant in discontinuous regions. The third-order TVD Runge–Kutta method is used for temporal discretization. Many one-dimensional and two-dimensional numerical examples show that the new scheme has the properties of high accuracy, low dissipation, and good shock-capturing ability.

A new type of third-order finite volume multi-resolution WENO schemes on tetrahedral meshes

In this continuing paper of Zhu and Shu (2018) [62], Zhu and Shu (2019) [63], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes Zhang and Shu (2009) [56] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.

A high-order nodal discontinuous Galerkin method for solution of compressible non-cavitating and cavitating flows

In this work, a high-order nodal discontinuous Galerkin method is applied and assessed for the simulation of compressible non-cavitating and cavitating flows. The one-fluid approach with the thermal effects is used to properly model the cavitation phenomenon. Here, the spatial and temporal derivatives in the system of governing equations are discretized using the nodal discontinuous Galerkin method and the third-order TVD Runge–Kutta method, respectively. Various numerical fluxes such as the Roe, Rusanov, HLL, HLLC and AUSM+-up and two discontinuity capturing methods, namely, the generalized MUSCL limiter and a generalized exponential filter are implemented in the solution algorithm. At first, the sinusoidal density wave problem which has a smooth solution is simulated and the effects of the numerical fluxes on the accuracy and performance of the nodal discontinuous Galerkin method are studied. Two problems, namely, the shock–density interaction (non-cavitating flow) and the two symmetric expansion waves (cavitating flow) are then computed and the effects of the numerical fluxes and the discontinuity capturing methods on the accuracy and computational cost of the solution are investigated. For non-cavitating flows, the high-pressure water–water shock tube and the low-pressure water–water shock tube are also simulated. Then, three cavitating flow problems, namely, the two symmetric expansion waves, the shock-condensation tube and the collapsing cavitation bubble are simulated to assess the accuracy and robustness of the solution algorithm. Results show that the solution methodology based on the high-order NDGM is accurate and robust for simulating the compressible non-cavitating and cavitating flows.

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

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.

Numerical simulation of viscoelastic fluid flows using a robust FVM framework on triangular grid

This paper presents a robust finite volume method (FVM) framework on unstructured triangular grid for simulating the viscoelastic flow of Oldroyd-B fluid. In order to enhance the numerical stability at high Weissenberg numbers, the framework adopts the CUBISTA scheme and the flux-based finite volume scheme to implement the spatial discretization of the Oldroyd-B constitutive equations, and the third-order TVD Runge–Kutta method to perform the temporal discretization. Meanwhile, we use the cell centered FVM to discretize the conservation equations of mass and momentum, and the IDEAL algorithm to solve the discrete equations for the sake of robustness and efficiency. The validity of the presented FVM framework is testified by comparing the numerical results with the analytical solutions for the simple two-dimensional (2D) channel flow. In addition, we also simulate two more complex viscoelastic problems, i.e., the 2D lid-driven cavity flow and the 2D abrupt 4:1 planar contraction flow. The calculated results demonstrate that our method has a good numerical stability and works well for simulating viscoelastic fluid flows at moderately high values of Weissenberg number.

High-resolution high-order upwind compact scheme-based numerical computation of natural convection flows in a square cavity

In this paper, a higher-order accuracy method is proposed for the solution of time-dependent nature convection problems based on the stream function-vorticity form of Navier–Stokes equations, in which an optimized third-order upwind compact scheme (Opt-UCD3) with high resolution is proposed to approximate the nonlinear convective terms, the fourth-order symmetrical Padé compact scheme is utilized to discretize the viscous terms, the fourth-order compact scheme on the nine-point 2D stencil is used for approximating the stream-function Poisson-type equation and the third-order TVD Runge–Kutta method is employed for the time discretization. To assess numerical capability of the newly proposed algorithm, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Moreover, the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are also computed for the wide range of Rayleigh numbers (103<Ra<1010). The characteristic parameters such as Nusselt number, velocity, and streamline show excellent agreement with benchmark solutions and some accurate results available in the literature. Additionally, the detailed features of flow phenomena for the higher Rayleigh numbers (108<Ra<1010) are delineated. The results show that the natural convection flow looses stability firstly via a Hopf bifurcation at Rac1 to the periodic flow regime, and then undergoes second bifurcation at a critical Rayleigh number Rac2 to quasi-periodic flow regime, and eventually transits to turbulent through a further bifurcation. In the periodic regime, there exist at least two branches of solutions. All of the results are agree well with ones in the literature and show the capabilities of the present method to properly simulate the unsteady nature convection problems.

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

A total variation diminishing-weighted average flux (TVD-WAF)-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL) Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL). The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.

High Resolution Numerical Simulation of Detonation Diffraction of Condensed Explosives

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.

Solving seven-equation model for compressible two-phase flow using multiple GPUs

In this paper, the application of an HLLC-type approximate Riemann solver in conjunction with the third-order TVD Runge–Kutta method to the seven-equation compressible two-phase model on multiple Graphics Processing Units (GPUs) is presented. Based on the idea proposed by Abgrall et al. that “a multiphase flow, uniform in pressure and velocity at t=0, will remain uniform on the same variables during time evolution”, discretization schemes for the non-conservative terms and for the volume fraction evolution equation are derived in accordance with the HLLC solver used for the conservative terms. To attain high temporal accuracy, the third-order TVD Runge–Kutta method is implemented in conjunction with operator splitting technique, in which the sequence of operators is recorded in order to compute free surface problems robustly. For large scale simulations, the numerical method is implemented using MPI/Pthread-CUDA parallelization paradigm for multiple GPUs. Domain decomposition method is used to distribute data to different GPUs, parallel computation inside a GPU is accomplished using CUDA, and communication between GPUs is performed via MPI or Pthread. Efficient data structure and GPU memory usage are employed to maintain high memory bandwidth of the device, while a special procedure is designed to synchronize thread blocks so as to reduce frequencies of kernel launching. Numerical tests against several one- and two-dimensional compressible two-phase flow problems with high density and high pressure ratios demonstrate that the present method is accurate and robust. The timing tests show that the overall speedup of one NVIDIA Tesla C2075 GPU is 31× compared with one Intel Xeon Westmere 5675 CPU core, and nearly 70% parallel efficiency can be obtained when using 8 GPUs.

High accuracy numerical investigation of double-diffusive convection in a rectangular enclosure with horizontal temperature and concentration gradients

Double-diffusive convection of a binary mixed fluid in a rectangular enclosure with horizontal temperature and concentration gradients is numerically investigated. The problem with thermal Rayleigh numbers of 104 and 105, the Prandtl number being range of [0.015,12], the Lewis number being range of [0.05,100], the buoyancy ratio of 0.8, and the height-to-width aspect ratio of 2 are considered. A compact difference method with fully fourth-order accuracy and high resolution in space, involving a, at least, fourth-order upwind compact scheme suggested for approximation of the nonlinear convective terms, a fourth-order symmetrical compact scheme to discretize the viscous terms and the third-order TVD Runge–Kutta method employed for time discretization, is proposed for solving this unsteady double-diffusive convection problem. The effects of Prandtl and Lewis numbers on flow structure, the temperature and concentration distribution are investigated and discussed for Ra=104 and 105 respectively in a rectangular enclosure with the aspect ratio A=2. A bifurcation for the double-diffusive convection is captured with the variation of Prandtl number, and two different types of periodic flow motion with Lewis number in different ranges are observed.

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

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.

Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves

Improvements in the numerical algorithm for the dynamics of flows that involve discontinuities and broadband fluctuations simultaneously are proposed. These two flow features suggest numerical strategies of a paradoxical nature because the discontinuities demand dissipation, and the small-scale smooth features require the opposite. There may be several ways to approach such a complicated issue, but the natural choice is a numerical technique that can adjust adaptively with flow regimes. The weighted essentially non-oscillatory (WENO) scheme may be this choice. However, there are two sources of dissipation associated with the WENO procedure: the upwind optimal stencil and the nonlinear adaption mechanism. The current work suggests a robust and comprehensive treatment for the minimization of dissipation error from these two sources. The optimization technique, which is guided by restriction on the linear optimal weights derived from stability and consistency requirement, is used to delay the dissipation of the upwind optimal stencil to those wavenumbers for which the dispersion error is large. The parallel advantage of this technique is the improvement of the dispersion property. Nevertheless, optimization decreases the formal order of accuracy of the optimal stencil from fifth order to third order. This loss of accuracy is derived by Taylor series expansion. Using Taylor-series expansion and WENO procedure, the third-order accuracy is verified in the smooth region, except at the critical point of order two, where the order of accuracy reduces to at least second order. This possible loss of accuracy at the second-order critical point is restored in an attempt to reduce the dissipation induced by the nonlinear adaptive weights. Modification of the nonlinear weights to reduce the dissipation is introduced by redefining them with an additional smoothness indicator. Other suggestions to minimize the dissipation of the nonlinear weights are also reviewed. The numerical approximation of the spatial derivative is performed by means of a conservative and consistent finite difference method based on monotone local Lax–Friedrichs Riemann solver. The resulting scheme is then integrated by the optimal third-order TVD Runge–Kutta method to ensure the nonlinear stability of the overall numerical method. A variety of benchmark problems, ranging from non-broadband to broadband, are solved using the proposed schemes and compared with the existing ones. Most test problems are validated against exact or reference data. The numerical results with bandwidth optimization and modification of the nonlinear weights are consistently superior.

Numerical Study of Compressible Mixing Layers Using High-Order WENO Schemes

This paper reports high resolution simulations using fifth-order weighted essentially non-oscillatory (WENO) schemes with a third-order TVD Runge-Kutta method to examine the features of turbulent m...

APPLICATION OF ADAPTIVE MESH REFINEMENT IN NUMERICAL SIMULATION OF GAS DETONATION

In this paper the two-dimensional Euler equations, with a simple chemical reaction model, are used as the governing equations for the detonation problem. The spatial derivatives are evaluated using the fifth-order WENO scheme, and the third-order TVD Runge-Kutta method is employed for the temporal derivative. The characteristics of the two-dimensional detonation in an argon-diluted mixture of hydrogen and oxygen are investigated using Adaptive Mesh Refinement (AMR) method. From computational accuracy point of view, AMR enables the detonation front to be clearer than the method with basic meshes. From the other point of computational time, AMR also saves about half the time as compared with the case of refining the entire field. It is obvious that AMR not only increases the resolution of local field, but also improves the efficiency of numerical simulation.