Classical Weighted Essentially Non-oscillatory Schemes Research Articles

A new high order hybrid WENO scheme for hyperbolic conservation laws

AbstractThis article proposes an improved hybrid weighted essentially non‐oscillatory (WENO) scheme based on the third‐ and fifth‐order finite‐difference modified WENO (MWENO) schemes developed by Zhu et al. in (SIAM J. Sci. Comput. 39 (2017), A1089–A1113.) for solving hyperbolic conservation laws. The MWENO schemes give a guideline on whether to use the WENO scheme or the linear upwind scheme. Unfortunately, because there is no explicit formula for computing the roots of algebraic polynomials of order four or higher, it is difficult to generalize this criterion to higher order cases. Therefore, this article proposes a simple criterion for constructing a series of seventh‐, ninth‐, and higher‐order hybrid WENO schemes, and then designs a class of improved smooth indicator WENO (WENO‐MS) schemes. Compared with the classical WENO schemes, the main advantages of the WENO‐MS schemes are their robustness and efficiency. And these WENO‐MS schemes are more efficient, have better resolution, and can solve many extreme problems without any additional techniques. Furthermore, a simplification criterion is proposed to further improve the computational efficiency of the WENO‐MS schemes, and these simple WENO‐MS schemes are abbreviated as WENO‐SMS schemes in this article. Extensive numerical results demonstrate the good performance of the WENO‐MS schemes and the WENO‐SMS schemes.

A new family of downwind-limited, scale-invariant WENO schemes with optimal accuracy

Despite improvement over decades, there are still at least two key issues not fully solved in Weighted Essentially Non-oscillatory (WENO) schemes. Firstly, most WENO schemes hardly satisfy the scale-invariant property, which undermines their robustness and accuracy when simulating flows involving different scales such as reactive multi-component flows and plasma. Secondly, most of existing Essentially Non-oscillatory (ENO) constraints are not sufficient and necessary conditions to achieve optimal accuracy. Therefore, in this work, we design a new family of WENO schemes based on a three-cell stencil to address these two issues. The new WENO non-linear weights are designed as bounded asymptotic functions which are dependent on cell Normalised Volume Integrated Average (NVIA) values and a shape parameter θ. Thus, the new WENO weighting functions are scale-independent. The shape parameter θ controls the errors between the nonlinear weight and the ideal weight. As the value of θ increases, the new weighting functions become more accurate but also become more likely to produce numerical oscillations. Thus, we propose a downwind limiting method as a supplementary constraint of ENO property. The maximum value of θ which corresponds to the optimal accuracy of a weighting function can then be determined. The accuracy analysis via the standard Taylor expansion at critical points reveals that the proposed weighting functions outperform classical WENO schemes. The accuracy, scale-invariant property and ENO property of proposed schemes are further validated through benchmark tests, which shows the superior performance of the new schemes in comparison with existing three-cell-based WENO schemes. Thus, this work proposes a new direction to construct scale-invariant WENO weighting functions with optimal accuracy, which can potentially be extended to higher-order WENO schemes.

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.

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.

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper, we develop two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain nonlinear high order derivatives, and possibly peakon solutions or shock waves. By introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic system, and the \mdp equation as a first order system. Then we choose a linear finite difference scheme with suitable order of accuracy for the auxiliary variable(s), and two finite difference WENO schemes with unequal-sized sub-stencils for the primal variable. One WENO scheme uses one large stencil and several smaller stencils, and the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil, both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights. Another advantage is that the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

Improved fifth-order weighted essentially non-oscillatory scheme with low dissipation and high resolution for compressible flows

Compressible flows are numerically simulated using hyperbolic conservation laws. This study proposes a modified fifth-order weighted essentially non-oscillatory (WENO) scheme with a relatively low dissipation and high resolution for hyperbolic conservation laws. This scheme exhibits good performance when solving complex compressible flow fields containing strong discontinuities and smooth microstructures. A simple local smoothness indicator and an eighth-order global smoothness indicator are introduced to improve the accuracy. Furthermore, we construct a new optimal coefficient, which can be adaptively adjusted with different states of the flow field. It no longer depends on the grid spacing. This adaptive coefficient not only reduces dissipation while improving the resolution but also prevents negative dissipation and effectively suppresses spurious numerical oscillations. The proposed scheme attains a higher accuracy at high-order critical points than three classical WENO schemes. Moreover, analysis of the approximate dispersion relation indicates that the proposed scheme provides good dispersion and dissipation properties compared with other WENO schemes. Finally, several standard numerical experiments are performed to demonstrate the enhanced performance of the proposed scheme. The numerical results indicate that the present scheme has a low dissipation, high resolution, and good stability to capture both smooth and discontinuous structures.

Multi-level WENO schemes with an adaptive characteristic-wise reconstruction for system of Euler equations

The Weighted Essentially Non-Oscillatory (WENO) scheme is an accurate and robust reconstruction procedure to simulate compressible flows, especially in the presence of discontinuities in the solution. The recently introduced multi-level WENO schemes like WENO-ZQ (Zhu & Qiu, 2016), WENO-AO (Balsara et al. 2016), WENO-ZS (Zhu & Shu 2019) provide better resolution of the solution near discontinuities and accuracy for smooth problems than classical WENO scheme. Despite having better accuracy, a multi-level WENO scheme is not computationally efficient as it involves the computation of an extra smoothness indicator as compared to the classical WENO scheme. In this article, we have proposed two new algorithms of multi-level schemes for system of Euler equations, which retains the accuracy and resolution of multi-level schemes while being less costly. The first algorithm involves identifying the troubled-cells and employing the characteristic-wise WENO reconstruction in the troubled-cells to avoid spurious oscillations and component-wise WENO reconstruction in smooth regions. While in the second algorithm, the latter part is replaced by the component-wise finite difference upwind reconstruction. The important aspect of the new algorithm is the troubled-cell indicator, which has been developed utilizing the information of the extra smoothness indicator of multi-level schemes. The present algorithms are independent of order of multi-level WENO schemes. To demonstrate the new idea, we have considered fifth order WENO-AO(5,3), WENO-ZQ, and WENO-ZS schemes. Several 1D & 2D numerical experiments are performed to demonstrate the efficiency of proposed algorithms and its performance is compared with multi-level schemes. The proposed algorithms are efficient and take 15%–75% less computational time than multi-level schemes while retaining the advantageous features of multi-level schemes.

A new fifth order finite difference WENO scheme to improve convergence rate at critical points

In the present work, we construct a new, improved version of fifth-order finite-difference Weighted Essentially Non-Oscillatory (WENO) scheme with less dissipation to approximate solutions for one- and two-dimensional hyperbolic conservation laws and associated problems. The higher-order information presented for the classical WENO scheme is applied to construct a new global smoothness indicator, a composition of first derivatives of local smoothness indicators with their respective interpolation polynomials. The sufficient condition of nonlinear weights is satisfied by using Taylor expansions. We verify the proposed scheme through several benchmark test cases for scalar and system of hyperbolic conservation laws. We observe from the numerical simulations that the designed WENO scheme demonstrates high resolutions, fifth-order accuracy, and robustness at critical points as compared with other modern WENO schemes.

Three-level order-adaptive weighted essentially non-oscillatory schemes

Classical fifth-order weighted essentially non-oscillatory (WENO) schemes are based on reconstructions from three consecutive second-order polynomials. They give a third-order accurate scheme when two consecutive polynomials are smooth and one is non-smooth. This situation means losing one order of accuracy if compared with a fourth-order scheme. In classical WENO schemes, weighting functions that determine the weights for the polynomials and that eventually define the reconstruction are directly related to a smoothness indicator. This property may lead to a non-zero residual of a very small value in the Taylor expansion. To handle this situation a weighting procedure that does not directly rely on the smoothness indicator is presented. Once the smooth polynomials are determined, the smoothness indicator has no role in the further steps, which are taken care of by a new switch function. Contrary to other approaches this switch function does not involve any conditional statements. A tuning parameter is also included so that the resulting order-adaptive property can be adjusted to specific requirements. The performance of the resulting schemes for cell-average and point-value reconstructions is studied. In addition, the effect of the choice of numerical flux or Riemann solver on these reconstruction strategies is analysed. Numerical tests for the Euler equations of gas dynamics in one and two space dimensions are presented. Improvements in resolution and the stability are found when compared with the conventional WENO techniques.

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations

In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order $$\alpha $$ $$(1<\alpha \le 2)$$ into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

A low dissipation finite difference nested multi-resolution WENO scheme for Euler/Navier-Stokes equations

In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller L1 and L∞ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.

A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations

In this paper, a new fifth-order finite difference well-balanced multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve for one-dimensional and two-dimensional shallow water equations with or without source terms on structured meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation once again. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure developed by Xing and Shu (2005) for balancing the flux gradients and the source terms, and then the new fifth-order well-balanced multi-resolution WENO scheme (Zhu and Shu, 2018) could satisfy the exact C-property for still stationary solutions, maintain the fifth-order accuracy in smooth regions, and keep essentially non-oscillatory property in non-smooth regions. Compared with the classical well-balanced WENO schemes (Xing and Shu, 2005), the new features of this finite difference well-balanced multi-resolution WENO scheme is its simplicity and hierarchical structure in obtaining higher-order accuracy and its linear weights could be arbitrarily chosen with one requirement that their summation is one. It is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing finite difference well-balanced WENO scheme for solving shallow water equations. This new well-balanced WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Some benchmark numerical examples are performed to illustrate the good performances of this new well-balanced WENO scheme which could get high-order accuracy, keep exact C-property, sustain good convergence property, and obtain sharp shock transitions in the whole computational field.

L1‐type smoothness indicators based weighted essentially nonoscillatory scheme for Hamilton‐Jacobi equations

SummaryThis article presents an improved fifth‐order finite difference weighted essentially nonoscillatory (WENO) scheme to solve Hamilton‐Jacobi equations. A new type of nonlinear weights is introduced with the construction of local smoothness indicators on each local stencil that are measured with the help of generalized undivided differences in L1‐norm. A novel global smoothness measurement is also constructed with the help of local measurements from its linear combination. Numerical experiments are conducted in one‐ and two‐dimensions to demonstrate the performance enhancement, resolution power, numerical accuracy for the proposed scheme, and compared it with the classical WENO scheme.

A new 6th-order WENO scheme with modified stencils

In this article, a new 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. As with previous 6th-order central-upwind WENO schemes, the present scheme is a convex combination of four candidate linear reconstructions. The difference is that the most upwind and downwind stencils use four cell values, while the inner two stencils nominally use three cell values but the original quadratic reconstructions are modified to be 4th-order approximations by adding cubic correction terms involving the five cell values of the classical 5th-order WENO scheme. Sixth-order accuracy of the new scheme in smooth regions including critical points is achieved by using a reference smoothness indicator. Several numerical examples show that the new scheme has higher resolution compared with the recently developed 6th-order WENO schemes.

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.

High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions

In this paper, a simple and efficient third-order weighted essentially non-oscillatory (WENO) reconstruction is developed for three-dimensional flows, in which the idea of two-dimensional WENO-AO scheme on unstructured meshes [42] is adopted. In the classical finite volume type WENO schemes, the linear weights for candidate stencils are obtained by solving linear systems at Gaussian quadrature points of cell interface. For the three-dimensional scheme, such operations at twenty-four Gaussian quadrature points of a hexahedron would reduce the efficiency greatly, especially for the moving-mesh computation. Another drawback of the classical WENO schemes is the appearance of negative weights with irregular local topology, which affect the robustness of spatial reconstruction. For the three-dimensional WENO-AO scheme, a simple strategy of selecting big stencil and sub-stencils for reconstruction is proposed. With the reconstructed quadratic polynomial from big stencil and linear polynomials from sub-stencils, the linear weights are chosen as positive numbers with the requirement that their summation equals to one and be independent of local mesh topology. With such WENO reconstruction, a high-order gas-kinetic scheme (HGKS) is developed for both three-dimensional inviscid and viscous flows. Taken the grid velocity into account, the scheme is extended into the moving-mesh computation as well. Numerical results are provided to illustrate the good performance of such new finite volume WENO scheme. In the future, such WENO reconstruction will be extended to the unstructured meshes.

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory (WENO) schemes (Zhu and Shu in J Comput Phys, 375: 659–683, 2018) is applied to solve for steady-state problems on structured meshes. Since the classical WENO schemes (Jiang and Shu in J Comput Phys, 126: 202–228, 1996) might suffer from slight post-shock oscillations (which are responsible for the residue to hang at a truncation error level), this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations. This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes, could obtain fifth-order, seventh-order, and ninth-order in smooth regions, and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO schemes 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-difference and finite-volume WENO schemes for solving steady-state problems. In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes, the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy

In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25,45], could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.

Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes

In this paper, we continue our research on the numerical study of convergence to steady-state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in Zhu and Shu (J Comput Phys 349:80–96, 2017), from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in Zhu and Qiu (SIAM J Sci Comput 40(2):A903–A928, 2018) for the third- and fourth-order versions. In this paper, we extend the design to a new fifth-order version in the same framework to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in Zhu and Shu (2017), by performing such spatial reconstruction procedures together with a TVD Runge–Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations that are responsible for the phenomenon wherein the residues of classical WENO schemes hang at a truncation error level instead of converging to machine zero. The third-, fourth-, and fifth-order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady-state simulations in our extensive numerical experiments.