Finite Difference WENO Research Articles

Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-conservative Products

Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-conservative Products

Numerical simulation of 3-D seismic wave based on alternative flux finite-difference WENO scheme

SUMMARY High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

Local Characteristic Decomposition–Free High-Order Finite Difference WENO Schemes for Hyperbolic Systems Endowed with a Coordinate System of Riemann Invariants

Local Characteristic Decomposition–Free High-Order Finite Difference WENO Schemes for Hyperbolic Systems Endowed with a Coordinate System of Riemann Invariants

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

A new type of increasingly higher order finite difference and finite volume MR-WENO schemes with adaptive linear weights for hyperbolic conservation laws

In this paper, the new fifth-order, seventh-order, and ninth-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) schemes with adaptive linear weights are presented for hyperbolic conservation laws on structured meshes. They are termed as high-order finite difference and finite volume ALW-WENO schemes. These ALW-WENO schemes only apply one small stencil and one large stencil in reconstruction processes, which could achieve the desired accuracy in the region of smoothness and non-oscillatory properties in the region of containing strong shocks. The linear weights that sum to one can be automatically adjusted to any positive numbers. This is the first time that arbitrary high-order finite difference and finite volume WENO schemes are designed by using only two unequal-sized central spatial stencils. The structure of these novel WENO schemes is simple, so it is easier for obtaining high-order accuracy and solving multi-dimensional problems in large scale engineering applications. Compared to traditional MR-WENO schemes with same order, the computational efficiency can be further improved. Some benchmark tests indicate that these new ALW-WENO schemes have good robustness and performance.

A sixth-order finite difference WENO scheme for non-linear degenerate parabolic equations: an alternative technique

PurposeThis paper aims to construct a sixth-order weighted essentially nonoscillatory scheme for simulating the nonlinear degenerate parabolic equations in a finite difference framework.Design/methodology/approachTo design this scheme, we approximate the second derivative in these equations in a different way, which of course is still in a conservative form. In this way, unlike the common practice of reconstruction, the approximation of the derivatives of odd order is needed to develop the numerical flux.FindingsThe results obtained by the new scheme produce less error compared to the results of other schemes in the literature that are recently developed for the nonlinear degenerate parabolic equations while requiring less computational times.Originality/valueThis research develops a new weighted essentially nonoscillatory scheme for solving the nonlinear degenerate parabolic equations in multidimensional space. Besides, any selection of the constants (sum equals one is the only requirement for them), named the linear weights, will obtain the desired accuracy.

High order WENO finite difference scheme with adaptive dual order ideal weights for hyperbolic conservation laws

An adaptive even order weighted essentially non-oscillatory polynomial reconstruction procedure (WPR) with Z-type weights (WENO-Za) is proposed to reduce the numerical dissipation and improve the resolution of the fine-scale structures in a long-time simulation of the hyperbolic conservation laws. The new spatial solver combines the even and odd order (even order plus one) WENO finite difference schemes together. The novel WENO-Za scheme adapts the ideal weights, instead of the nonlinear weights, between the even and odd order based on a nonlinear transition function, which is a function of the odd order global smoothness indicator and a user-adjustable parameter. The dissipation and dispersion behaviors of the WENO-Za scheme is illustrated by the approximate dispersion relation technique. For the Euler equations, the WPR procedure is performed on each positive and negative split flux variable in the characteristic space. The performance of the WENO-Za scheme, in terms of resolution, essentially non-oscillatory shock-capturing, and efficiency, are evaluated by solving several benchmark shock-tube problems, such as the extended shock-entropy interaction problem, the extended shock-density interaction problem, the two blast-waves interaction problem, the Riemann problem, and the forward-facing step problem. The results show that the WENO-Za scheme can 1) retain the fine-scale structures in the smooth regions better due to the reduction of numerical dissipation of the even order scheme in exchange for a slightly larger dispersive error in a long-time simulation and 2) capture localized strong gradients and shocks in an essentially non-oscillatory manner due to the strong sub-stencil biasing in the odd order scheme. The space-time evolutions of the transition functions demonstrate the robust adaptive nature of the WENO-Z4a5 scheme in the regions of shock, contact discontinuity, and rarefaction waves.

Fifth order AWENO finite difference scheme with adaptive numerical diffusion for Euler equations

In solving hyperbolic conservation laws using the fifth-order characteristic-wise alternative WENO finite-difference scheme (AWENO) with Z-type affine-invariant nonlinear Ai-weights, the classical local Lax–Friedrichs flux (LLF) is modified with an adaptive numerical diffusion (ND) coefficient to form an adaptive LLF flux (LLF-M). The adaptive ND coefficient depends nonlinearly on the local scale-independent smoothness measures on the pressure, density, dilation, and vorticity. The feature sensor combines the well-known Durcos’ sensor on the dilation and vorticity of the velocity field and Jameson’s sensor on the density and pressure. Based on the measure of the feature sensor, the ND coefficient of the LLF-M flux is designed to transit smoothly and quickly from a set minimum to the maximum, the local spectral radius of the eigenvalues of the Jacobian of the flux. Hence, the modified AWENO scheme improves the resolution of small-scale structures due to a substantial reduction of excessive dissipation while capturing discontinuities essentially non-oscillatory (ENO-property). The performance of the improved AWENO scheme is validated in one- and two-dimensional benchmark problems with discontinuities and smooth small-scale structures. The results show that the new adaptive LLF-M flux improves resolution, captures fine-scale structures, and is robust in a long-term simulation.

A New Third-Order Finite Difference WENO Scheme to Improve Convergence Rate at Critical Points

In this work, a new, improved third-order finite difference weighted essentially non-oscillatory scheme is presented for one- and two-dimensional hyperbolic conservation laws and associated problems. The parameter p which is regulate dissipation is introduced in the nonlinear weights in the framework of the conventional WENO-Z scheme, and the higher-order global smoothness indicator is obtained by the idea of Wang [Wang, Y. H., Y. L. Du, K. L. Zhao and L. Yuan. 2020. ‘A Low-dissipation Third-order Weighted Essentially Nonoscillatory Scheme with a New Reference Smoothness Indicator’. International Journal for Numerical Methods in Fluids. 92 (9): 1212–1234.], the sufficient condition of nonlinear weights is proved by using Taylor expansions. Finally, the value range of parameter p is obtained. The proposed scheme is verified to achieve the optimal order near critical points by linear convection equations with different initial values, and the high-resolution characteristic of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.

Finite difference alternative unequal-sized weighted essentially non-oscillatory schemes for hyperbolic conservation laws

In this paper, two unequal-sized weighted essentially non-oscillatory (US-WENO) schemes are proposed for solving hyperbolic conservation laws. First, an alternative US-WENO (AUS-WENO) scheme based directly on the values of conserved variables at the grid points is designed. This scheme can inherit all the advantages of the original US-WENO scheme [J. Zhu and J. Qiu, “A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws,” J. Comput. Phys. 318, 110–121 (2016).], such as the arbitrariness of the linear weights. Moreover, this presented AUS-WENO scheme enables any monotone fluxes applicable to this framework, whereas the original US-WENO scheme is only suitable for the more dissipative smooth flux splitting. Therefore, the method in this paper has a smaller L1 and L∞ numerical errors than the original scheme under the same conditions. Second, in order to further improve the computational efficiency of the above AUS-WENO scheme, a hybrid AUS-WENO scheme is proposed by combining a hybrid strategy. This strategy identifies the discontinuous regions directly based on the extreme points of the reconstruction polynomial corresponding to the five-point stencil, which brings the important advantage that it does not depend on the specific problem and does not contain any artificial adjustable parameters. Finally, the performance of the above two AUS-WENO schemes in terms of low dissipation, shock capture capability, discontinuity detection capability, and computational efficiency is verified by some benchmark one- and two-dimensional numerical examples.

A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two-dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (Z. Zhao et al. (2020) [27]), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient, and has better resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and is more efficient than the same order finite difference WENO scheme of Jiang and Shu (1996) [11] and WENO scheme with artificial linear weights of Zhu and Qiu (2016) [29]. Various benchmark numerical examples are presented to show the fifth-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

A new finite difference mapped unequal-sized WENO scheme for Hamilton-Jacobi equations

In this paper, a new fifth-order finite difference mapped unequal-sized weighted essentially non-oscillatory (MUS-WENO) scheme is proposed for solving Hamilton-Jacobi equations in multi-dimensions. This new MUS-WENO scheme uses the same widest spatial stencils as that of the classical same order finite difference WENO scheme [19], and could obtain smaller truncation errors and optimal fifth-order accuracy in smooth regions. The MUS-WENO scheme uses a convex combination of a quartic polynomial with two quadratic polynomials. The linear weights can be artificially set as any positive constants on condition that their summation is one. Together with a new mapping function, associated new mapped nonlinear weights are proposed to reduce the difference with the linear weights. So it is a first time that the new finite difference MUS-WENO scheme with a very small ε could reach optimal accuracy order convergence even near extreme points in smooth regions and avoid spurious oscillations near strong discontinuities when solving Hamilton-Jacobi equations. Generally speaking, the main advantages of such new MUS-WENO scheme comparing with the classical WENO scheme [19] are its good convergence, robustness, efficiency and easy extension to multi-dimensions. Numerical experiments are proposed to show the good performance of this new MUS-WENO scheme.

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the “lake equations” for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

High Order Semi-implicit WENO Schemes for All-Mach Full Euler System of Gas Dynamics

In this paper, we propose high order semi-implicit schemes for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. The schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes. Besides, they can well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

Numerical study of shock-disturbances interaction in hypersonic inviscid flows with real gas effects using high-order WENO scheme

In the present study, the shock-disturbances interaction in hypersonic inviscid flows with real gas effects is studied by applying a high-order accurate numerical method with the shock capturing technique. To consider real gas effects, the equilibrium air model is utilized here. The strong conservative form of the unsteady compressible Euler equations in the 2D generalized curvilinear coordinates is formulated and the resulting system of equations for the equilibrium air model is discretized by using the fifth-order finite-difference WENO scheme in space and the explicit third-order TVD Runge–Kutta scheme in time to provide a highly accurate and robust equilibrium airflow solver. The solution methodology adopted based on the high-order WENO scheme with the shock capturing technique can reasonably simulate the strong discontinuities in the solution domain without needing any numerical artifact and this feature makes the solution method suitable for the shock-disturbances interaction in hypersonic flows in which real gas effects are to be considered. To study the physical phenomenon of the shock-disturbances interaction problem in high-speed inviscid equilibrium airflows, two test cases are simulated by applying the solution methodology adopted. In the first test case, the interaction between the incoming fast acoustic wave with the bow shock formed around a cylinder with the nose radius of RN=0.0381 m at M∞=8.03 and T∞=182.333 K is simulated. In the second test case, the cylinder radius is taken RN=2.54×10−2 m and the upstream velocity and temperature are u∞=5590m/s and T∞=1833 K, respectively, and all kinds of free stream disturbances, including the entropy wave and the fast and slow acoustic waves, for different wave numbers are considered. The computations are carried out for both the perfect gas and equilibrium air models and the effects of real gas on both the mean and disturbances fields are also studied. The present study introduces the first-known numerical investigation of the shock-disturbances interaction in hypersonic flow over blunt noses with real gas effects in which the numerical results obtained are thoroughly validated by the analytical/theoretical ones and good agreement is found.

Simple high order well-balanced finite difference WENO schemes for the Euler equations under gravitational fields

The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite difference weighted essentially non-oscillatory (WENO-R/I) schemes, which base on the WENO reconstruction and interpolation procedures respectively, for this system with an ideal gas equation of state and a prescribed gravitational field. The main idea in achieving the well-balanced property is to rewrite the source terms in a special way and discretize them by using the nonlinear WENO differential operators with the homogenization condition to those for the flux terms. The proposed well-balanced schemes can be proved mathematically to preserve the hydrostatic isothermal and polytropic equilibria states exactly and at the same time maintain genuine high order accuracy. Moreover, the resulting schemes are high efficiency of computation and can be implemented straightforwardly into the existing finite difference WENO code with the Lax-Friedrichs numerical flux for solving the compressible Euler equations. Last but not least, the proposed well-balanced framework also works for a class of moving equilibrium state. Extensive one- and two-dimensional numerical examples are carried out to investigate the performance of the proposed schemes in term of high order accuracy, well-balanced property, shock capturing essentially non-oscillatory and resolving the small perturbation on the coarse mesh resolution.

A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws

In this paper, we develop a modified fifth order accuracy finite difference Hermite WENO (HWENO) scheme for solving hyperbolic conservation laws. The main idea is that we first modify the derivatives of the solution by Hermite WENO interpolations, then we discretize the original and derivative equations in the spatial directions by the same approximation polynomials. Comparing with the original finite difference HWENO scheme of Liu and Qiu (J Sci Comput 63:548–572, 2015), one of the advantages is that the modified HWENO scheme is more robust than the original one since we do not need to use the additional positivity-preserving flux limiter methodology, and larger CFL number can be applied. Another advantage is that higher order numerical accuracy than the original scheme can be achieved for two-dimensional problems under the condition of using the same approximation stencil and information. Furthermore, the modified scheme preserves the nice property of compactness shared by HWENO schemes, i.e., only immediate neighbor information is needed in the reconstruction, and it has smaller numerical errors and higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu (J Comput Phys 126:202–228, 1996). Various benchmark numerical tests of both one-dimensional and two-dimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed novel HWENO scheme.

An improved WENO method based on Gauss-kriging reconstruction with an optimized hyper-parameter

An adaptive finite-difference WENO method with Gauss-kriging reconstruction (we call it WENO-K) is proposed to reduce dissipation in smooth regions of flow while preserving high-resolution around discontinuities for hyperbolic system of conservation laws. The method adopts a kriging model with non-polynomial Gauss exponential function to obtain new reconstruction coefficients that contain a hyper-parameter. By adaptively optimizing the hyper-parameter and automatically identifying troubled cells using newly developed indicators, the accuracy in the smooth region is obviously improved. Compared with the classical WENO-JS method, the proposed WENO-K method provides more accurate reconstructions and sharper solution profiles near discontinuities. Furthermore, the WENO-K method is easy to implement in an existing classical WENO code with less than 13%–16% of additional computational cost. Numerical results demonstrate that the proposed method outperforms the WENO-JS method for a broad range of problems. This method is supposed to be applied to other variants of WENO scheme and offers the potential of improving their accuracy.

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

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