Order Weighted Essentially Non-oscillatory Scheme Research Articles

Shock wave induced by the collapse of a bubble cluster with each bubble distributed randomly

The phenomenon of bubble collapse is ubiquitous in both natural and industrial settings, often manifesting in the form of bubble clusters. Despite its prevalence, there has been limited research on the dynamics of bubble clusters. In order to gain a better understanding of shock wave emission, liquid jet propagation, and the resulting pressure loads during bubble cluster collapse, we have developed a high-fidelity numerical approach aimed at accurately capturing shock waves and phase interfaces. The interface compression technique is introduced into the compressible two-phase model to improve the sharpness of the phase interface. We solve the physical model using the BVD (Boundary Variation Diminishing) principle. Both the 5th order WENO (Weighted Essentially Non-Oscillatory) scheme and the interface capturing function THINC (Tangent of Hyperbola for INterface Capturing) are utilized to reconstruct primary variables at cell boundaries, effectively minimizing numerical dissipation. For further resolution improvement, we implement these numerical methods on the AMR (Adaptive Mesh Refinement) mesh. Our numerical results demonstrate that the distance between the wall and the bubble cluster, as well as the nearest bubble to the wall, significantly influence the wall peak pressure. However, they have a minimal impact on the overall evolution of the bubble cluster and the peak pressure induced in the flow field. Remarkably, the spatial distribution of bubbles and the total number of bubbles play a crucial role in shaping the dynamic behavior of the bubble cluster and the pressure loads exerted both on the solid wall and in the flow field. The crucial findings offer profound insights into a range of applications, including cavitation erosion, underwater explosions, and drug delivery.

A new type of improved third order WENO scheme with finite difference framework

In this paper, we present a novel variety of third order finite difference weighted essentially non-oscillatory (WENO) scheme based on WENO-ZQ scheme to approximate the computational solutions of one- and two-dimensional non-linear hyperbolic conservation laws. The new reconstruction process of fluxes is the convex combination of a second degree polynomial and two linear polynomials. We apply a new modified global smoothness indicator into the non-linear weights. We also use the power parameter ‘p’ to balance the accuracy of the proposed scheme. This novel third order WENO scheme applies three points data {xi−1,xi,xi+1} same as WENO-JS3 and WENO-Z3 schemes. In the proposed scheme, the main advantages are that the associated linear weights can be taken as any positive numbers on condition their summation is one. Also, it is easy to be used in multi-dimensional engineering problems. We implement the time discretization by using third order accurate strong stability preserving Runge–Kutta method. The extensive computational experiments of a collection of benchmark test problems are provided to demonstrate high resolution, robustness and absolute truncation errors in L1 and L∞ norms. We observe from the results that the new scheme contains better capability in performance than WENO-PZ3 scheme.

A fifth order WENO scheme for numerical simulation of shallow granular two-phase flow model

In this article, a weighted essentially non-oscillatory (WENO) scheme is implemented to simulate two-phase shallow granular flow (TPSF) model. The flow is assumed to be incompressible and it is regarded as shallow layer of granular and liquid material. The mathematical model consists of two phases, that is, solid and liquid. Each phase has its continuity and momentum equation. The presence of the equations are coupled together involving the derivatives of unknowns which make it more challenging to solve. An efficient numerical technique is needed to tackle the numerical complexities. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model. The weighted essentially non-oscillatory (WENO) scheme of order 5 is utilized to handle the shock waves and contact discontinuities appear in the solution. The results are compared with the results already available in the literature by conservation element and solution element (CESE) scheme. It is observed the WENO scheme produces less errors as compared to CESE scheme and also effectively handle the shocks.

Preconditioned Jacobian-free Newton–Krylov fully implicit high order WENO schemes and flux limiter methods for two-phase flow models

Motivated by the high-resolution properties of high-order Weighted Essentially Non-Oscillatory (WENO) and flux limiter (FL) for steep-gradient problems and the robust convergence of Jacobian-free Newton-Krylov (JFNK) methods for nonlinear systems, the preconditioned JFNK fully implicit high-order WENO and FL schemes are proposed to solve the transient two-phase two-fluid models. Specially, the second-order fully-implicit BDF2 is used for the temporal operator and then the third-order WENO schemes and various flux limiters can be adopted to discrete the spatial operator. For the sake of the generalization of the finite-difference-based preconditioning acceleration methods and the excellent convergence to solve the complicated and various operational conditions, the random vector instead of the initial condition is skillfully chosen as the solving variables to obtain better sparsity pattern or more positions of non-zero elements in this paper. Finally, the WENO_JFNK and FL_JFNK codes are developed and then the two-phase steep-gradient problem, phase appearance/disappearance problem, U-tube problem and linear advection problem are tested to analyze the convergence, computational cost and efficiency in detailed. Numerical results show that WENO_JFNK and FL_JFNK can significantly reduce numerical diffusion and obtain better solutions than traditional methods. WENO_JFNK gives more stable and accurate solutions than FL_JFNK for the test problems and the proposed finite-difference-based preconditioning acceleration methods based on the random vector can significantly improve the convergence speed and efficiency.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Very high order WENO schemes using efficient smoothness indicators

Very high order weighted essentially non-oscillatory (WENO) schemes can obtain more accurate results than those relatively low order WENO schemes for many problems. However, as the order increases, the candidate sub-stencils of WENO schemes will contain more node points. Then, the classical smoothness indicators become very complex and need much more floating point operations. Although they could be written in a more succinct form which is a sum of perfect squares, only a little improvement of the computational efficiency is obtained. Furthermore, for the power parameter p in the nonlinear weights of the previous (2r-1)th order WENO schemes, only the range of choices is given and the optimal value could be problem dependent. In this article, another class of very high order WENO schemes is constructed by using more efficient smoothness indicators. Compared with very high order WENO schemes using classical smoothness indicators, the new ones are more time efficient, and less error prone for programming. Furthermore, they have the same approximate dispersion relation (ADR) as their underlying linear schemes. And a more important advantage is that they have both good behavior near critical points and good stability near discontinuities. For Euler equations, a predetermined order reduction technique including two inequalities is proposed to deal with the problems caused by interactions between discontinuities of different characteristic fields. One inequality is constructed using this kind of smoothness indicators of pressure to suppress the non-physical oscillations. And the other is proposed according to the governing equation of the pressure to prevent the emergence of negative pressure. Several test problems are presented to demonstrate the good properties of the proposed WENO schemes.

A friction interface model for multi-material interactions in a Eulerian framework

A friction model for hyper-elastic solid materials was proposed for the simulation of complex shear impacts in a Eulerian framework. The interfacial status of multi-material interactions was obtained in Harten, Lax, and van Leer discontinuity (HLLD) Riemann solver. The inverse deformation gradient tensor was utilized in governing equations to describe the shape change of hyper-elastic solid materials. The level set method (LSM) was used to identify multiple materials in computational domain and the interface boundary condition was applied using the real ghost fluid method (RGFM). A uniform eigensystem of hyperbolic equations was derived using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO) scheme. Six numerical tests of one-dimensional problems were used to illustrate the robustness of the 5th order WENO scheme. The proposed friction model was evaluated for impact cases in both one and two dimensions with three-dimensional components. The redistribution of strain and kinetic energy during solid-solid interactions was represented using the proposed friction model, while both ‘slip’ and ‘stick’ interface showed unphysical approximations. We believe that the friction interface model completes the theory of multi-material interaction and provide a more appropriate way to model the complex dynamic behavior in Eulerian framework for solid materials.

High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations

A numerical framework of the generalized form of high order well-balanced finite difference weighted essentially non-oscillatory (WENO) interpolation-based schemes is proposed for the shallow water equations. It demonstrates more flexible construction process than the classical WENO reconstruction-based schemes. The weighted compact nonlinear schemes and finite difference alternative WENO schemes are two specific cases. To maintain the exact C-property, the splitting technique for the source term in the finite difference scheme [Xing and Shu, J. Comput. Phys. 208 (2005)] and the reconstruction technique in the finite volume WENO scheme [Xing and Shu, J. Comput. Phys. 214 (2006)] are adopted. The proposed scheme can be proved mathematically to maintain the exact C-property and demonstrates numerically that it is well-balanced by construction for the stationary water surface. Moreover, the local characteristic projections are employed to further mitigate the Gibbs oscillations. The proposed generic high order WENO schemes not only achieve high order accuracy but also capture the high gradients/shock waves essentially non-oscillatory. Meanwhile, the small perturbation problems can be resolved well on a coarse grid.

An Improvement of Third Order WENO Scheme for Convergence Rate at Critical Points with New Non-linear Weights

In this paper, we construct and implement a new improvement of third order weighted essentially non-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws. In our approach, a modification in the global smoothness measurement is reported by applying all three points on global stencil $$(i-1,i,i+1)$$ which is used for convergence of non-linear weights towards the optimal weights at critical points and achieves the desired order of accuracy for third order WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numerical accuracy in smooth regions. The computational performance of the proposed WENO scheme with this global smoothness measurement is verified in several benchmark one- and two-dimensional test cases for scalar and vector hyperbolic equations. Extensive computational results confirm that the new proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 and WENO-F3 schemes.

Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes

A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50]) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23]) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws

A new type of finite difference weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws was designed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), in this continuing paper, we extend such methods to finite volume version in multi-dimensions. There are two major advantages of the new WENO schemes superior to the classical finite volume WENO schemes (Shu, in: Quarteroni (ed) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer, Berlin, 1998), the first is the associated linear weights can be any positive numbers with only requirement that their summation equals one, and the second is their simplicity and easy extension to multi-dimensions in engineering applications. The new WENO reconstruction is a convex combination of a fourth degree polynomial with two linear polynomials defined on unequal size spatial stencils in a traditional WENO fashion. These new fifth order WENO schemes use the same number of cell average information as the classical fifth order WENO schemes Shu (1998), could get less absolute numerical errors than the classical same order WENO schemes, and compress nonphysical oscillations nearby strong shocks or contact discontinuities. Some benchmark tests are performed to illustrate the capability of these schemes.

A new fifth order finite difference WENO scheme for Hamilton‐Jacobi equations

AbstractIn this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non‐oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton‐Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1095–1113, 2017

An efficient adaptive high-order scheme based on the WENO process

In this paper, we propose an efficient adaptive scheme based on the WENO (weighted essentially non-oscillatory) process for the hyperbolic conservation laws. This scheme achieves fifth order accuracy as the fifth order WENO scheme does. Also, due to its adaptive mechanism, the scheme deals with problems which contain both discontinuities and complex smooth regions much better than traditional high order schemes. In addition, the proposed scheme saves computational costs by avoiding solving redundant nonlinear weights in certain places. Systematic analyses and numerical tests show that our adaptive scheme is of a high level of robustness and resolution.

Numerical simulation of wave transformation, breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations (FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory (WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations (NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento (Italy).

A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws

In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme 14,20, could get less absolute truncation errors in L 1 and L ∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme 14,20 in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.

An improved mapped weighted essentially non-oscillatory scheme

In this paper we develop an improved version of the mapped weighted essentially non-oscillatory (WENO) method in Henrick et al. (2005) [10] for hyperbolic partial differential equations. By rewriting and making a change to the original mapping function, a new type of mapping functions is obtained. There are two parameters, namely A and k, in the new mapping functions (see Eq. (13)). By choosing k=2 and A=1, it leads to the mapping function in Henrick et al., i.e.; the mapped WENO method by Henrick et al. actually belongs to the family of our improved mapped WENO schemes. Furthermore, we show that, when the new mapping function is applied to any (2r-1)th order WENO scheme for proper choice of k, it can achieve the optimal order of accuracy near critical points. Note that, if only one mapping is used, the mapped WENO method by Henrick et al., whose order is higher than five, can not achieve the optimal order of accuracy in some cases. Through extensive numerical tests, we draw the conclusion that, the mapping function proposed by Henrick et al. is not the best choice for the parameter A. A new mapping function is then selected and provides an improved mapped WENO method with less dissipation and higher resolution.

Weights Design For Maximal Order WENO Schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200---212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202---228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191---3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.

Maximum-principle-satisfying High Order Finite Volume Weighted Essentially Nonoscillatory Schemes for Convection-diffusion Equations

To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] to convection diffusion equations, we propose a nonconventional high order finite volume weighted essentially nonoscillatory (WENO) scheme which can be proved maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.

Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes

The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202---228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the "essentially non-oscillatory" property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput. 31:273---305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.

Fast Sweeping Fifth Order WENO Scheme for Static Hamilton-Jacobi Equations with Accurate Boundary Treatment

A fifth order weighted essentially non-oscillatory (WENO) fast sweeping method is designed in this paper, extending the result of the third order WENO fast sweeping method in J. Sci. Comput. 29, 25---56 (2006) and utilizing the two approaches of accurate inflow boundary condition treatment in J. Comput. Math. 26, 1---11 (2008), which allows the usage of Cartesian meshes regardless of the domain boundary shape. The resulting method is tested on a variety of problems to demonstrate its good performance and CPU time efficiency when compared with lower order fast sweeping methods.