Weighted Essentially Non-oscillatory Research Articles

A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order

Discretization of the convection term in a three-cell compact stencil has been widely used in Computational Fluid Dynamics (CFD) of engineering because the second-order interpolation scheme is considered to be more robust and algorithmically simple than very high-order schemes. However, according to Godnunov's theorem, it has never been trivial to design an accurate and oscillation-free discretization scheme in a compact stencil. Over decades, various kinds of schemes have been proposed to increase accuracy beyond second-order without producing non-physical numerical oscillations. Representative schemes are polynomial-based non-linear interpolations such as TVD (Total Variation Diminishing), NVD (Normalized Variable Diagram) and third-order WENO (Weighted Essentially Non-oscillatory); and non-polynomial-based interpolations including RBF (Radial Basis Functions), LDLR (Local Double Logarithmic Reconstruction) and Tangent of Hyperbola for INterface Capturing (THINC). Within the semi-discretization finite volume method, existing reconstruction schemes suffer from different issues to some extent. For example, existing TVD schemes tend to cause smearing, clipping and squaring effects. Most WENO schemes and inconsistent RBF schemes are not scale-invariant. Especially, the improved WENO-Z scheme is neither scale-invariant nor mesh-size-independent. Computationally expensive non-polynomial-based interpolations usually have singularity and are dependent on tunable parameters. Thus, this work reviews reconstruction schemes by reformulating or calculating the reconstructed boundary value in the normalised-variable space where the characteristics of reconstruction schemes can be read directly. Through the accuracy analysis and the induction of existing schemes, this work proposes a Unified Normalised-variable Diagram (UND) where the following properties of a reconstruction scheme can be given diagrammatically: 1) the second-order and the third-order accuracy condition, 2) the TVD and CBC (Convection Boundedness Criterion) constrain, 3) the ENO condition, 4) the High Resolution (HR) property, 5) and the numerical dissipation and anti-dissipation error.The proposed UND framework is firstly demonstrated by the improvement of the existing schemes with a shape-preserving TVD limiter and a scale-invariant WENO scheme. Then, by defining the normalised reconstruction operator, a brand-new scheme named ROUND (Reconstruction Operator on Unified Normalised-variable Diagram) is proposed. The ROUND scheme is devised by directly configuring the normalised reconstruction operator of favourable properties on UND. This work presents a dissipative ROUND scheme in which normalised reconstruction operators are fully in the dissipative region of UND, and a low-dissipative ROUND scheme where rigorously adjusted anti-dissipation errors are introduced. Different formulations of ROUND schemes are presented to show the flexibility of designing schemes within the UND framework. The proposed dissipative and low-dissipative ROUND schemes are evaluated via benchmark tests which show the superior performance of ROUND on accuracy and resolution. Especially, in some cases, the low-dissipative ROUND schemes obtain comparable and even better resolution than classic fifth-order WENO. The significance of this work is further demonstrated by the preliminary results of applying ROUND on the FDM-IBM (Finite Difference Method and Immersed Boundary Method) framework.

Nonlinear improvements of quasi-interpolanting splines to approximate piecewise smooth functions

Quasi-interpolation based on B-spline approximation methods are used in numerous applications. However, we observe that the Gibbs phenomenon appears when approximating near discontinuities. We present nonlinear modifications, based on weighted essentially non-oscillatory (WENO) techniques, of well-known quasi-interpolation methods to avoid this phenomena near discontinuities and, at the same time, maintain the high-order accuracy in smooth regions.

Three-dimensional third-order gas-kinetic scheme on hybrid unstructured meshes for Euler and Navier–Stokes equations

In this paper, a third-order gas-kinetic scheme is developed on the three-dimensional hybrid unstructured meshes for the compressible inviscid and viscous flows. In the classical weighted essentially non-oscillatory (WENO) scheme, the high-order spatial accuracy is achieved by the non-linear combination of lower order polynomials. However, for the hybrid unstructured meshes, the procedures, including the selection of candidate stencils and calculation of linear weights, become extremely complicated, especially for three-dimensional problems. To overcome the drawbacks, a third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A general strategy is given for the selection of big stencil and sub-stencils for hybrid meshes, and the topologically independent linear weights are used in the spatial reconstruction. A unified interpolation is also used for the volume integral of different types of control volumes, as well as the flux integration over different cell interfaces. Incorporate with the two-stage fourth-order temporal discretization, the explicit high-order gas-kinetic schemes are developed for unsteady problems. With the lower–upper symmetric Gauss–Seidel (LU-SGS) methods and Jacobi iteration, the implicit high-order gas-kinetic schemes are developed for steady problems. To accelerate the computation, both explicit and implicit scheme is implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). Especially, for implicit parallel computation, Jacobi iteration is used. The speedup of GPU code suggests the potential of high-order gas-kinetic schemes for the large scale computation. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.

An efficient finite difference IFWENO-THINC hybrid scheme for capturing discontinuities

The weighted essentially non-oscillatory (WENO) scheme is a famous shock-capturing scheme, but for high-precision numerical simulation, the classical WENO scheme is always considered to have excessive dissipation and insufficient resolution near discontinuities. Improving the accuracy order or reducing the dissipation of the WENO by optimizing the linear weights can effectively improve the resolution of this scheme in the smooth regions. However, under the same method, the improvement of shock-capturing abilities is generally limited and may lead to numerical oscillations. In order to simultaneously improve the resolution in discontinuous regions and ensure the performance of the smooth regions, this study uses the shock detector to combine the tangent of hyperbola for interface capturing (THINC) scheme, which is a better discontinuity-capturing scheme, with a given WENO discretization, such as the improved fast WENO (IFWENO) scheme that equips with high economy and does not loss accuracy at any critical point. The hybrid method in this study is an apriori algorithm, which is more efficient and universal than the method in (Takagi et al. J. Comput. Phys. 452(2022)110899). Additionally, the performance of the THINC scheme in different regions near the discontinuity is analyzed in detail through numerical experiments, and a shock detector that can fully exploit the advantages of the THINC scheme is provided. By using this shock detector, several one-dimensional and two-dimensional numerical cases are employed to verify that the hybrid scheme can greatly improve the performance in discontinuities without affecting the flow structure in the smooth regions, which proves the effectiveness of the proposed shock detector and the excellence of the hybrid IFWENO-THINC scheme.

Nonlinear weights for shock capturing schemes with unconditionally optimal high order

We develop in this paper nonlinear weights for shock capturing schemes, such that the design optimal high order is achieved regardless of any order of the critical point. In particular, we validate the proposed nonlinear weights by testing their performance for a fifth-order weighted compact nonlinear scheme (WCNS), which has advantages in terms of implementing numerical fluxes and keeping the freestream property on curvilinear grids. Theoretical analysis is performed to demonstrate the unconditionally optimal high order of the proposed method. In addition, the spectral property of the proposed nonlinear weights is compared with some existing nonlinear ones. Several one- and two-dimensional benchmark examples are presented to validate the advantages of the proposed method. It is worth noting that the developed nonlinear weights can be applied straightforwardly for widely used weighted essentially non-oscillatory (WENO) schemes.

A sufficient condition for free-stream preserving WENO schemes on curvilinear grids of complex geometries

The well developed conservative finite-difference-method based on upwind schemes, such as the weighted essentially nonoscillatory (WENO) schemes, may violate the free-stream preserving (FP) property when applied to curvilinear grids due to the loss of the geometric conservation law. Recent developments in FP techniques have disregarded the balance between the computational accuracy and efficiency, and more importantly, the general regularity for the FP property is not available. In this paper, we rigorously prove that the upwind dissipation satisfies the FP property as long as the sub-stencil reconstructions of the metrics are identical by rearranging the dissipation part as an accumulative form composed of the sub-stencil reconstructions. With this, the novel FP metrics are constructed straightforwardly for various WENO reconstructions, introducing less complexity to implement the schemes with a merely simple accuracy compensation by replacing the central fluxes with high-order ones. Applications of such an implementation to the isotropic vortex convection and double Mach problem etc. validate the accuracy and robustness of the novel FP method in simulating both the smooth flows and discontinuities at lower computational costs than the previous methods. Besides, further viscous computations for the three-element airfoil and the 3D wing are conducted to show the good capability of the present method on complex geometries.

Order enhanced finite volume methods through non-polynomial approximation

In this paper, we introduce an approximation method that establishes certain order enhancements by leveraging radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at cell interfaces. The aforementioned improvement in the order of accuracy is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters appearing in the basis. This paper seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as a hybrid implementation. To highlight the effectiveness of the non-polynomial basis in shock-capturing, the proposed methods are applied to systems of one-dimensional hyperbolic and weakly hyperbolic conservation laws and compared with several well-known WENO schemes in the literature. We also include a two-dimensional example for a scalar problem that demonstrates an extension to multiple dimensions. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The numerical results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. A key contribution of this work is the development of robust third-order WENO method, which further demonstrates the effectiveness of the non-polynomial basis.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.

Investigations on a high order SPH scheme using WENO reconstruction

Theoretically, a 2nd order convergence can be reached with the Smoothed Particle Hydrodynamics (SPH) method. However, depending on the spatial disorder of particles, the order of convergence observed in practice can be lower than one. In this paper, a methodology is proposed for the reconstruction of high order numerical fluxes in Riemann-SPH formulations for weakly-compressible flows, so as to increase the global order of convergence of the scheme. This methodology is based on the use of one-dimensional Weighted Essentially Non-Oscillatory (WENO) reconstructions applied at each pair of interacting particles, and each 1D WENO stencil is completed using Moving-Least-Squares (MLS) reconstructions. It is shown that a 6th order convergence can be reached with the proposed SPH-WENO scheme. The gain in accuracy and convergence properties of this scheme is shown and discussed through its application to various one-dimensional and two-dimensional test cases. In particular, the influence of the number of neighbor particles and of the spatial particle disorder is studied. Finally, it is shown that the proposed high-order SPH scheme provides a better accuracy to CPU time ratio than usual Riemann-SPH schemes. The treatment of boundary conditions on rigid walls with the proposed scheme is also discussed.

Evaluation of finite difference based asynchronous partial differential equations solver for reacting flows

Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical methods, scalable algorithms and re-design of current state-of-the art numerical solvers are required for scaling to these machines with minimal overheads. One such approach for partial differential equations based solvers involves computation of spatial derivatives with possibly delayed or asynchronous data using high-order asynchrony-tolerant (AT) schemes to facilitate mitigation of communication and synchronization bottlenecks without affecting the numerical accuracy. In the present study, an effective methodology of implementing temporal discretization using a multi-stage Runge-Kutta method with AT schemes is presented. Together these schemes are used to perform asynchronous simulations of canonical reacting flow problems, demonstrated in one-dimension including auto-ignition of a premixture, premixed flame propagation and non-premixed autoignition. Simulation results show that the AT schemes incur very small numerical errors in all key quantities of interest including stiff intermediate species despite delayed data at processing element (PE) boundaries. For simulations of supersonic flows, the degraded numerical accuracy of well-known shock-resolving WENO (weighted essentially non-oscillatory) schemes when used with relaxed synchronization is also discussed. To overcome this loss of accuracy, high-order AT-WENO schemes are derived and tested on linear and non-linear equations. Finally the novel AT-WENO schemes are demonstrated in the propagation of a detonation wave with delays at PE boundaries.

ENO-ET: a reconstruction scheme based on extended ENO stencil and truncated highest-order term

High-order numerical methods for hyperbolic balance laws must circumvent Godunov’s theorem, for which non-linear spatial reconstruction is a successful procedure; this allows the design of non-linear semi-discrete and fully discrete high-order finite volume methods. The Essentially-Non-Oscillatory (ENO) method pioneered by Harten and collaborators in 1987 was a fundamental breakthrough in providing the bases for constructing schemes of both high-order of accuracy in smooth regions and essentially non-oscillatory at discontinuities. Several variations of ENO have since appeared in the literature, notably the Weighted Essentially-Non-Oscillatory (WENO) method of Jiang and Shu. In the present article, after revisiting the main elements of reconstruction methods, including their analysis, we propose a new, conservative ENO-type reconstruction procedure, called ENO-ET, that preserves the main advantages of the classical ENO and successfully overcomes its well-known weaknesses. The newly proposed reconstruction scheme is systematically assessed and compared with the best existing reconstruction schemes. We do so in two ways: (i) the methods are purely regarded as local, non-linear interpolation objects and (ii) the reconstruction methods are implemented in the context of high-order fully-discrete ADER numerical schemes for solving hyperbolic balance laws. The resulting schemes are of arbitrary order of accuracy in both space and time. In this paper the methods are implemented up to seventh order of accuracy and tested on carefully chosen numerical examples. Tests solved include the linear advection equation, the Euler equations of gas dynamics and the blood flow equations. Particular features of interest here are: (a) attaining theoretically expected convergence rates for smooth solutions arising from demanding initial conditions, (b) compute essentially non-oscillatory profiles for discontinuous solutions and (c) attain successfully steady-steady state solutions by time marching. As compared to existing methods, the newly proposed ENO-ET scheme performs very well for all these problems.

A high order positivity-preserving polynomial projection remapping method

In this paper, we develop a high-order positivity-preserving polynomial projection remapping method based on the L2 projection for the discontinuous Galerkin (DG) scheme. Combined with the Lagrangian type DG scheme and the rezoning strategies, we present an indirect arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method. By clipping precisely the intersections between the old distorted mesh and the new rezoned mesh, our remapping method is high-order accurate and has no limitation for the mesh movements, so it is suitable for the large deformable problems. A positivity-preserving limiter is also added for the physical variables in computational fluid dynamics without losing the original high-order accuracy and conservation. A multi-resolution weighted essentially non-oscillatory (WENO) limiter is adopted to overcome numerical oscillations and it can keep the original high-order accuracy in the smooth region. This WENO limiter combines several different degrees of polynomials which are the local L2 projections of the original polynomial with nonlinear weights calculated by their smoothness, therefore, it is highly parallel efficient. The properties of positivity-preserving, non-oscillation and high-order accuracy of the remapping method will be shown by a variety of numerical experiments on one, two and three dimensional unstructured meshes. The performance of the ALE-DG scheme with rezoning and remapping is also tested for the Euler system in one and two dimensions.

A hybrid physics-informed neural network for nonlinear partial differential equation

The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous solutions. In this paper, we focus on the discrete time physics-informed neural network (PINN), and propose a hybrid PINN (hPINN) scheme for the nonlinear PDEs. In this approach, the local solution structures are classified as smooth and nonsmooth scales by introducing a discontinuity indicator, and then the automatic differentiation technique is employed for resolving smooth scales, while an improved weighted essentially nonoscillatory (WENO) scheme is adopted to capture discontinuities. We then test the present approach by considering the viscous and inviscid Burgers equations, and it is shown that compared with original discrete time PINN, the present hPINN approach has a better performance in approximating the discontinuous solution even at a relatively larger time step.

A class of high-order improved fast weighted essentially non-oscillatory schemes for achieving optimal order at any critical points

The weighted essentially non-oscillatory (WENO) scheme is one of the most popular shock-capturing schemes, and constructing a more efficient and higher-order WENO scheme has always been an intention of optimization design. In the general WENO reconstruction framework, the smoothness indicator plays an important role in identifying whether the sub-stencils are in discontinuous or smooth regions. However, the classical smoothness indicator is the most expensive one in the whole reconstruction algorithm, and its computational complexity increases sharply with the improvement of the accuracy order. Therefore, a class of efficient and superior WENO schemes called improved fast WENO (IFWENO) are proposed based on the fast WENO (FWENO). To improve efficiency, the smoothness indicator of the IFWENO scheme is simplified from the traditional version, and the nonlinear weight calculation method is modified. The parameter ε is carefully designed to obtain the superior property that the accuracy of the spatial derivatives will not degrade at any order critical point in smooth regions. The reason for the instability occurring in the high-order FWENO is revealed, and the parameter p is likewise specifically selected to improve robustness at discontinuities. The excellent multi-scale resolution of the proposed IFWENO scheme is proven by theoretical analyses and numerical experiments. Through several typical examples, the consistently high accuracy and efficiency of the designed scheme in both smooth and discontinuous regions are verified.

Efficient implementation of high-order WENO schemes with sharing function for solving Euler equations

Due to the high-order accuracy and essentially non-oscillatory (ENO) property, the weighted ENO (WENO) schemes have a wide range of successful applications. The component-wise reconstruction WENO (CP WENO) scheme for fluxes or variables has simple formulations but it may produce numerical oscillations near discontinuities when solving the Euler equations. Although the characteristic-wise reconstruction WENO (CH WENO) scheme can reduce such oscillations, it involves too many characteristic projection operations. In this paper, first, we introduced a sharing function to indicate the discontinuities in the Euler equations and then constructed new adaptive characteristic-wise WENO (Ada-WENO) scheme and common-weights WENO (Co-WENO) scheme with this function. Several one and two dimensional problems are used to test the performances of Ada-WENO and Co-WENO. Numerical results show that, Ada-WENO can reduce the computational cost of CH WENO while maintaining its oscillation-free property since it only switches from CP WENO to CH WENO near discontinuities, and Co-WENO can reduce the cost and oscillations of CP WENO, but it may still generate few oscillations.

A numerical approach to a 2D porous-medium mathematical model: Application to an atherosclerosis problem

This work deals with the study and implementation of a Local Space-Time DG ADER approach, in the finite volume framework with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space, in the context of 2D porous media problems. The particular application performed concerns a biomedical problem dealing with the first stages of atherosclerosis disease, where the artery is considered as a porous medium, although the methodology described can be applied to other situations. The mathematical model on which this research is based is given by a system of two-dimensional nonlinear reaction-diffusion equations with a nonlinear source term in one of the equations, which is a variant of the model proposed originally in N. El Khatib, S. Genieys & V. Volpert (2007) Math. Model. Nat. Phenom., vol 2(2), pp. 126–141 [1]. The 2D model considered in this work has two main features: (i) the artery is taken as a porous medium and (ii) a nonlinear non-homogeneous Neumann boundary condition is incorporated, aimed to account for the recruitment of immune cells through the upper boundary, as a response to the production of cytokines. Certain theoretical properties of the stationary solutions and the evolution solution are stated and proved. Some of them are also verified with the numerical simulation results.

Structure-preserving finite volume arbitrary Lagrangian-Eulerian WENO schemes for the shallow water equations

This paper develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshes. The rigorous theoretical analysis and numerical examples demonstrate the high order accuracy and positivity-preserving property of the schemes under the ALE framework. For the well-balanced schemes, it is successful in the unique exact equilibrium preservation and capturing small perturbations of the hydrostatic state well without numerical oscillations near the discontinuity. Moreover, our ALE-WENO hybrid schemes have an advantage over the simulations on static meshes due to the higher resolution interface tracking of the fluid motion.

A high order positivity-preserving conservative WENO remapping method based on a moving mesh solver

In this paper, a high order accurate positivity-preserving conservative remapping algorithm is developed. Quadrilateral meshes in two dimensions are used as examples. This remapping method is based on the numerical solution of the trivial equation ∂u∂t=0 on a moving mesh, which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=T. A high order finite volume scheme on the moving mesh is used to solve this problem. Specifically, we adopt the multi-resolution weighted essentially non-oscillatory (WENO) method for the spatial discretization and a strong stability preserving (SSP) Runge-Kutta method for the temporal discretization. The remapping algorithm is high order accurate under very mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which can always be satisfied with a suitable choice of the final pseudo-time T. Furthermore, we design our remapping algorithm to have positivity-preserving property by using the linear scaling positivity-preserving limiter so that the algorithm could ensure the positivity-preserving property of relevant physical variables and maintain conservation and original order of accuracy. A series of numerical experiments are given to demonstrate the properties of our remapping algorithm such as high order accuracy, essentially non-oscillatory performance, positivity-preserving and high computational efficiency.

A modified high-order symmetrical WENO scheme for hyperbolic conservation laws

Abstract This paper designs a modified weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws. Using the switching principle based on inflection points, the new scheme automatically adapts between linear upwind and WENO schemes. If there is at least one inflection point in the largest stencil available for reconstruction, a symmetrical WENO (SWENO) scheme is considered for the reconstruction of the numerical flux; otherwise the numerical flux is directly approximated by the reconstruction polynomial. By comparing the new scheme introduced in this paper with the classical WENO scheme and another improved scheme that has been proposed recently D. Chai, G. Xi, Z. Sun, Z. Wangand Z. Huang,Comput. Fluids, vol. 170, pp. 176–186, 2018), we can point out the robustness and better efficiency of this scheme. To examine and explain the features of the new scheme, a number of examples such as Euler equations have been considered.

A new hybrid CRWENO-MWENOZ-adaptive moving mesh method for cavitating two-phase compressible fluid flow simulation

In this paper, a high-order hybrid method for solving compressible two-phase fluid flow, including cavitation, is presented. In this regard, assuming pressure and temperature equilibrium, mass and heat transfer between the different phases are modeled. In this work, the CRMWENOZ method, which is a new combination of compact and weighted essentially non-oscillatory (WENO) methods and is more accurate than conventional methods, is presented. The new high-order hybrid method aims to predict the density and the pressure discontinuities in two-phase flow by combining the CRMWENOZ high-order method and an adaptive moving mesh technique. For this purpose, the adaptive moving mesh partial differential equation (MMPDE) method would also improve the accuracy of results by concentrating on the grid nodes in high-gradient regions for transient flows. Applying the CRMWENOZ method alongside MMPDE and using the fifth-order Radau method for time discretization lead to a substantial improvement in the accuracy of simulation, particularly near the liquid–gas interfaces. The accuracy of the proposed hybrid method was compared to other studies’ predictions of one-dimensional (1D) expansion and shock tube problems containing two-phase flow with and without cavitation. The results showed that the hybrid method presented was more accurate than the usual two-phase flow methods while using a reasonable amount of computer resources.