Third-order Total Variation Diminishing Runge Kutta Research Articles

A high-order simulation method for compressible multiphase flows with condensed-phase explosive detonation in underwater explosions

This study introduces an efficient method designed for the simulation of compressible multiphase flows associated with explosive detonation, primarily in the context of underwater explosions. The proposed approach integrates three core components: the compressible Euler equations, the level-set equation, and the program burn model. Spatial terms of the compressible Euler equations undergo discretization using a fifth-order accuracy weighted essentially non-oscillation reconstruction, while the third-order total variation diminishing Runge–Kutta scheme manages the temporal terms. The level-set method ensures accurate tracking of the multiphase interface. To detail the transition from solid, non-reactive explosives to gaseous detonation products in the condensed charge's detonation reaction zone, the program burn model based on Zeldovich, von Neumann, Doering theory. The efficacy and accuracy of the incorporated program burn model and the multiphase interface capture method are employed through four benchmark tests, exhibiting excellent agreement with previously published data from alternative numerical methods or commercial software. In conclusion, applying the proposed method to four distinct engineering scenarios facilitates a comprehensive understanding of the inherent dynamics associated with detonation and shock wave generation and propagation.

Nonlinear transient analysis of rigid rotor mounted on externally pressurized double-layered porous gas journal bearings accounting velocity slip

The present work aims at estimating the journal center trajectories of a rigid rotor mounted on a hybrid, finite gas lubricated journal bearings with double-layered porous bushing using nonlinear transient analysis. To consider velocity slip in the film at the interface between film and porous region, Reynolds equation is modified using the Beavers–Joseph boundary condition. The governing equations of flow at clearance, porous regions are discretized using finite volume method. The variable values at cell-face centers are obtained using interpolation scheme of third order. Those discretized equations coupled with equations of rigid rotor motion are solved by third-order total variation diminishing Runge–Kutta scheme. Influence of velocity slip and design parameters on critical mass parameter values were explored. The observations are presented in the form of graphs that serves as a reference during the design of such bearings.

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.

A numerical dispersion-dissipation analysis of discontinuous Galerkin methods based on quadrilateral and triangular elements

The dispersive and dissipative properties of numerical methods are important for numerical modeling. We have evaluated a numerical dispersion-dissipation analysis for two discontinuous Galerkin methods (DGMs) — the flux-based DGM (FDGM) and the interior penalty DGM (IP DGM) for scalar wave equation. The semidiscrete analysis based on the plane-wave analysis is conducted for quadrilateral and triangular elements. Two kinds of triangular elements are taken into account. The fully discrete analysis for each method is conducted by incorporating a classic third-order total variation diminishing (TVD) Runge-Kutta (RK) time discretization. Our results indicate that FDGM produces smaller numerical dispersion than IP DGM, but it introduces more numerical dissipation. Notably, the two methods have different local convergence orders for numerical dispersion and dissipation. The anisotropy properties for different mesh types can also be identified. Several numerical experiments are carried out that verify some theoretical findings. The experiments exhibit that the numerical error introduced by FDGM is less than that introduced by IP DGM whereas the storage and calculation time of FDGM are greater than that of IP DGM. Overall, our work indicates that when both methods adopt third TVD RK time discretization, the computational efficiency of FDGM is slightly larger than IP DGM.

Variational multi-scale finite element method for the two-phase flow of polymer melt filling process

PurposeThe purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface.Design/methodology/approachThe algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time.FindingsThe numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems.Originality/valueFor the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

A direct time-integral THINC scheme for sharp interfaces

This paper describes an improved multi-dimensional Tangent of Hyperbola INterface Capturing (THINC) scheme that uses a direct time integration of the hyperbolic tangent function. The new time-integral THINC scheme is based on the assumption that the interface remains unchanged and moves with the velocity of the cell center during a computational time step. A time-varying hyperbolic tangent function representing the two-phase distribution is then constructed and directly integrated in time. Compared with existing THINC methods that generally use third-order total variation diminishing Runge–Kutta (RK3) schemes to update the volume fraction, the proposed method requires only one reconstruction step, thus reducing the computational cost. Several classical advection tests have been implemented, and the results indicate that the proposed direct time-integral THINC method: 1) achieves computational errors close to those of the RK3 scheme at a computational cost that is close to that of the first-order explicit scheme; and 2) preserves better boundedness of the volume fractions than the original RK3-based THINC methods when using a larger steepness parameter β and higher Courant numbers. The proposed approach is employed with the incompressible Navier–Stokes equations to solve a 2D dambreak problem, and the numerical results agree well with the corresponding experimental data, demonstrating the applicability of the direct time-integral THINC method.

Low-Dissipative Hybrid Compressible Solver Designed for Large-Eddy Simulation of Supersonic Turbulent Flows

To reduce the numerical dissipation in turbulence modeling while maintaining the numerical stability around flow discontinuities in supersonic flowfield, a low-dissipative compressible solver is developed for large-eddy simulation within the OpenFOAM framework. To achieve the aforementioned goals, the low-dissipative solver adopts the hybrid scheme approach through combining the dissipative Kurganov–Tadmor scheme with the nondissipative central scheme via a shock sensor. In the construction of the central scheme, a robust skew-symmetric form of the convective term is adopted to preserve the local kinetic energy without adding an explicit dissipative term. Another feature of the low-dissipative solver is the implementation of an optimal explicit strong stability-preserving linear third-order total variation diminishing Runge–Kutta method for the temporal discretization. Numerical tests for a series of canonical flow problems are carried out to validate the solver’s good performance in the flowfield either with strong discontinuities or with continuous spectrum characteristics. Large-eddy simulation of a scramjet combustor with supersonic airstream passing over the flame holder is conducted to validate the low-dissipative solver’s reliability in a realistic flow with the complex interaction of shock discontinuities and turbulence.

An exponential time-integrator scheme for steady and unsteady inviscid flows

An exponential time-integrator scheme of second-order accuracy based on the predictor–corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge–Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

This paper is concerned with generalized polynomial chaos (gPC) approximation for first-order quasilinear hyperbolic systems with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Then the gPC stochastic Galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is discretized by a path-conservative finite volume WENO scheme in space and a third-order total variation diminishing Runge–Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional gPC Galerkin system is carried over via an operator splitting technique. Several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

A compressible multiphase flow model for violent aerated wave impact problems

This paper focuses on the numerical modelling of wave impact events under air entrapment and aeration effects. The underlying flow model treats the dispersed water wave as a compressible mixture of air and water with homogeneous material properties. The corresponding mathematical equations are based on a multiphase flow model which builds on the conservation laws of mass, momentum and energy as well as the gas-phase volume fraction advection equation. A high-order finite volume scheme based on monotone upstream-centred schemes for conservation law reconstruction is used to discretize the integral form of the governing equations. The numerical flux across a mesh cell face is estimated by means of the HLLC approximate Riemann solver. A third-order total variation diminishing Runge–Kutta scheme is adopted to obtain a time-accurate solution. The present model provides an effective way to deal with the compressibility of air and water–air mixtures. Several test cases have been calculated using the present approach, including a gravity-induced liquid piston, free drop of a water column in a closed tank, water–air shock tubes, slamming of a flat plate into still pure and aerated water and a plunging wave impact at a vertical wall. The obtained results agree well with experiments, exact solutions and other numerical computations. This demonstrates the potential of the current method to tackle more general wave–air–structure interaction problems.

A highly accurate method to solve Fisher’s equation

In this study, we present a new and very accurate numerical method to approximate the Fisher’s-type equations. Firstly, the spatial derivative in the proposed equation is approximated by a sixth-order compact finite difference (CFD6) scheme. Secondly, we solve the obtained system of differential equations using a third-order total variation diminishing Runge–Kutta (TVD-RK3) scheme. Numerical examples are given to illustrate the efficiency of the proposed method.

Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method

In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge–Kutta scheme in time have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems.

A compact finite difference method for the solution of the generalized Burgers–Fisher equation

AbstractIn this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compact finite difference method with minimal computational effort. To verify this, a combination of a sixth‐order compact finite difference scheme in space and a low‐storage third‐order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. The approximate solutions to the equation have been computed without transforming the equation and without using linearization. Comparisons indicate that there is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation

A numerical solution of the one-dimensional Burgers’ equation is obtained using a sixth-order compact finite difference method. To achieve this, a tridiagonal sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge–Kutta scheme in time have been combined. The scheme is implemented to solve two test problems with known exact solutions. Comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort. The present results are also seen to be more accurate than some available results given in the literature.

RAM: A Relativistic Adaptive Mesh Refinement Hydrodynamics Code

The authors have developed a new computer code, RAM, to solve the conservative equations of special relativistic hydrodynamics (SRHD) using adaptive mesh refinement (AMR) on parallel computers. They have implemented a characteristic-wise, finite difference, weighted essentially non-oscillatory (WENO) scheme using the full characteristic decomposition of the SRHD equations to achieve fifth-order accuracy in space. For time integration they use the method of lines with a third-order total variation diminishing (TVD) Runge-Kutta scheme. They have also implemented fourth and fifth order Runge-Kutta time integration schemes for comparison. The implementation of AMR and parallelization is based on the FLASH code. RAM is modular and includes the capability to easily swap hydrodynamics solvers, reconstruction methods and physics modules. In addition to WENO they have implemented a finite volume module with the piecewise parabolic method (PPM) for reconstruction and the modified Marquina approximate Riemann solver to work with TVD Runge-Kutta time integration. They examine the difficulty of accurately simulating shear flows in numerical relativistic hydrodynamics codes. They show that under-resolved simulations of simple test problems with transverse velocity components produce incorrect results and demonstrate the ability of RAM to correctly solve these problems. RAM has been tested in one, two and three dimensions andmore » in Cartesian, cylindrical and spherical coordinates. they have demonstrated fifth-order accuracy for WENO in one and two dimensions and performed detailed comparison with other schemes for which they show significantly lower convergence rates. Extensive testing is presented demonstrating the ability of RAM to address challenging open questions in relativistic astrophysics.« less

A Discontinuous Galerkin Global Shallow Water Model

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

The third-order relaxation schemes for hyperbolic conservation laws

A class of semi-discrete third-order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws. These schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing (TVD) in the zero relaxation limit. A third-order TVD Runge–Kutta splitting method is developed for the temporal discretization of the semi-discrete schemes. Numerical results are given illustrating these schemes on one-dimensional nonlinear problems.