Weighted Essentially Non-oscillatory Method Research Articles

Enhancing Accuracy in Numerical Simulations for High-Speed Flows: Integrating High-Order Corrections with Weighted Essentially Non-Oscillatory Flux

This study introduces a novel method to enhance numerical simulation accuracy for high-speed flows by refining the weighted essentially non-oscillatory (WENO) flux with higher-order corrections like the modified weighted compact scheme (MWCS). Numerical experiments demonstrate improved sharpness in capturing shock waves and stability in complex conditions like two interacting blast waves. Key highlights include simultaneous capture of small-scale smooth fluctuations and shock waves with precision surpassing the original WENO and MWCS methods. Despite the significantly improved accuracy, the extra computational cost brought by the new method is only marginally increased compared to the original WENO, and it outperforms MWCS in both accuracy and efficiency. Overall, this method enhances simulation fidelity and effectively balances accuracy and computational efficiency across various problems.

Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity–streamfunction (VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and O. Schilling, “A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,” Physica D 401, 132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer–Meshkov instability using a model of a Mach 1.3 air(acetone)/SF6 shock tube experiment [J. W. Jacobs and V. V. Krivets, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability,” Phys. Fluids 17, 034105 (2005)]. A comparison of the density fields from both simulations with the experimental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure. The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experiment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime. Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the VS and WENO results are in good agreement.

A new third-order EXP-WENO scheme for Hamilton–Jacobi equations

The authors of this research paper have introduced a new method for solving Hamilton–Jacobi (HJ) equations called third-order weighted essentially nonoscillatory (WENO) method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. The smoothness indicators are created using a finite difference operator that eliminates exponential polynomials. The numerical flux is constructed using an interpolation method based on exponential polynomials, which results in better outcomes around steep gradients. The new method maintains a high level of accuracy (i.e. three) in smooth regions, even near critical points. The authors have presented some numerical results to demonstrate the effectiveness of the new method and compared it with other WENO methods.

Path-conservative positivity-preserving well-balanced finite volume WENO method for porous shallow water equations

The present study constructed a path-conservative high-order positivity-preserving well-balanced finite volume Riemann solver for the one-dimensional porous shallow water equations with discontinuous porosity and bottom topography. First, the finite difference equations were formulated using the path-conservative approach to treat non-conservative products in source terms. As the solution depends on the paths, the family of paths was formulated for the solutions to satisfy the mass and energy conservations over the smooth paths connecting the discontinuous geometry (porosity and/or bottom topography). Such a property was achieved using the stationary wave, a stationary weak solution of the porous shallow water equations. Second, to increase computational accuracy, the weighted essentially non-oscillatory (WENO) and the Runge-Kutta methods were implemented for the spatiotemporal discretization of the porous shallow water equations, respectively. Third, a positivity-preserving limiter was employed to provide robustness and efficiency in handling wet and dry problems. Finally, all procedures in the present model were formulated to ensure the well-balanced property (i.e., the exact conservation of initial still water conditions over the irregular geometry). A set of numerical experiments was performed to verify that the present model exhibits positivity-preserving and well-balanced properties along with high-order accuracy and the shock-capturing ability for all types of Riemann problems, including eight wet cases and five dry cases. The numerical results of all cases agree with the analytical solutions.

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.

A physical-constraint-preserving finite volume WENO method for special relativistic hydrodynamics on unstructured meshes

This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten–Lax–van Leer flux, and third-order strong-stability-preserving time discretization. Due to the relativistic effects, the primitive variables (namely, the rest-mass density, velocity, and pressure) are highly nonlinear implicit functions in terms of the conservative variables, making the design and analysis of our method nontrivial. To address the difficulties arising from the strong nonlinearity, we adopt a novel quasilinear technique for the theoretical proof of the PCP property. Three provable convergence-guaranteed iterative algorithms are also introduced for the robust recovery of primitive quantities from admissible conservative variables. We also propose a slight modification to an existing WENO reconstruction to ensure the scaling invariance of the nonlinear weights and thus to accommodate the homogeneity of the evolution operator, leading to the advantages of the modified WENO reconstruction in resolving multi-scale wave structures. Extensive numerical examples are presented to demonstrate the robustness, expected accuracy, and high resolution of the proposed method.

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.

A hybrid WENO scheme for steady-state simulations of Euler equations

For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations.

A neural network enhanced weighted essentially non-oscillatory method for nonlinear degenerate parabolic equations

In this paper, a new modification of the weighted essentially non-oscillatory (WENO) method for solving nonlinear degenerate parabolic equations is developed using deep learning techniques. To this end, the smoothing indicators of an existing WENO algorithm, which are responsible for measuring the discontinuity of a numerical solution, are modified. This is done in such a way that the consistency and convergence of our new WENO-DS (deep smoothness) method is preserved and can be theoretically proved. A convolutional neural network (CNN) is used and a novel and effective training procedure is presented. Furthermore, it is shown that the WENO-DS method can be easily applied to additional dimensions without the need to retrain the CNN. Our results are presented using benchmark examples of nonlinear degenerate parabolic equations, such as the equation of a porous medium with the Barenblatt solution, the Buckley–Leverett equation, and their extensions in two-dimensional space. It is shown that in our experiments, the new method outperforms the standard WENO method, reliably handles sharp interfaces, and provides good resolution of discontinuities.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations

We develop and use a novel mixed-precision weighted essentially non-oscillatory (WENO) method for solving the Teukolsky equation, which arises when modeling perturbations of Kerr black holes. We show that WENO methods outperform higher-order finite-difference methods, standard in the discretization of the Teukolsky equation, due to the need to add dissipation for stability purposes in the latter. In particular, as the WENO scheme uses no additional dissipation, it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio binaries. In the mixed-precision approach, the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of \(\approx 3.3\). In addition, we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations. Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics. We apply our solver to study the behavior of the Aretakis charge—a conserved quantity, that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.

Hybrid high-resolution RBF-ENO method

Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. The RBF-ENO method is highly flexible in terms of geometry, but its stencil selection algorithm is computational expensive. In this work, we combine the computationally efficient WENO method and the geometrically flexible RBF-ENO method in a hybrid high-resolution essentially nonoscillatory method to solve hyperbolic conservation laws. The scheme is based on overlapping patches with ghost cells, the RBF-ENO method for unstructured patches and a standard WENO method on structured patches. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods to stabilize the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness and flexibility on benchmarks and complex test cases such as the scramjet inflow problem and a conical aerospike nozzle jet simulation.

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.

High Order Finite Difference Multi-resolution WENO Method for Nonlinear Degenerate Parabolic Equations

In this paper, we propose a new finite difference weighted essentially non-oscillatory (WENO) scheme for nonlinear degenerate parabolic equations which may contain non-smooth solutions. An alternative formulation is designed to approximate the second derivatives in a conservative form. In this formulation, the odd order derivatives at half points are used to construct the numerical flux, instead of the usual practice of reconstruction. Moreover, the multi-resolution WENO scheme is designed to circumvent the negative ideal weights and mapped nonlinear weights that appear when applying the standard WENO idea. We will describe the scheme formulation and present numerical tests for one- and two-dimensional, demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

Self-consistent quantum field theory for the characterization of complex random media by short laser pulses

We present a quantum field theoretical method for the characterization of disordered complex media with short laser pulses in an optical coherence tomography setup (OCT). We solve this scheme of coherent transport in space and time with weighted essentially nonoscillatory methods (WENO). WENO is preferentially used for the determination of highly nonlinear and discontinuous processes including interference effects and phase transitions like Anderson localization of light. The theory determines spatiotemporal characteristics of the scattering mean free path and the transmission cross section that are directly measurable in time-of-flight (ToF) and pump-probe experiments. The results are a measure of the coherence of multiple scattering photons in passive as well as in optically soft random media. Our theoretical results of ToF are instructive in spectral regions where material characteristics such as the scattering mean free path and the diffusion coefficient are methodologically almost insensitive to gain or absorption and to higher-order nonlinear effects. Our method is applicable to OCT and other advanced spectroscopy setups including samples of strongly scattering mono- and polydisperse complex nano- and microresonators.

On the reconstruction of discontinuous functions using multiquadric RBF–WENO local interpolation techniques

We discuss several approaches involving the reconstruction of discontinuous one-dimensional functions using parameter-dependent multiquadric radial basis function (MQ-RBF) local interpolants combined with weighted essentially non-oscillatory (WENO) techniques, both in the computation of the locally optimized shape parameter and in the combination of RBF interpolants. We examine the accuracy of the proposed reconstruction techniques in smooth regions and their ability to avoid Gibbs phenomena close to discontinuities. In this paper, we propose a true MQ-RBF–WENO method that does not revert to the classical polynomial WENO approximation near discontinuities, as opposed to what was proposed in Guo and Jung (2017) [12,13]. We present also some numerical examples that confirm the theoretical approximation orders derived in the paper.

Adaptive mapping for high order WENO methods

In this paper, a novel mapping approach through the use of adaptive mapping functions is introduced for high order weighted essentially non-oscillatory (WENO) methods. The new class of adaptive mapping functions are designed to adjust themselves to the solution based on a simple parameter calculated using the smoothness indicators that are readily available during computation. It is shown that this adaptive nature allows the resultant mapped WENO scheme to maintain sub-stencil weights close to the optimal weights in smooth regions without amplifying the weights of non-smooth stencils containing discontinuities. Therefore, adaptive mapping achieves enhanced accuracy in smooth regions and is more resistant against spurious oscillations near discontinuities. Taylor series analysis of the seventh order finite volume WENO scheme has been performed to demonstrate the loss of accuracy of the original WENO method near critical points. The convergence rates of the seventh order finite volume WENO scheme with adaptive mapping have been shown through a simple numerical example. Excellent results have been obtained for one-dimensional linear advection cases especially over long output times. Improved results have also been obtained for one- and two-dimensional Euler equation test cases.

A variable high-order shock-capturing finite difference method with GP-WENO

We present a new finite difference shock-capturing scheme for hyperbolic equations on static uniform grids. The method provides selectable high-order accuracy by employing a kernel-based Gaussian Process (GP) data prediction method which is an extension of the GP high-order method originally introduced in a finite volume framework by the same authors. The method interpolates Riemann states to high order, replacing the conventional polynomial interpolations with polynomial-free GP-based interpolations. For shocks and discontinuities, this GP interpolation scheme uses a nonlinear shock handling strategy similar to Weighted Essentially Non-Oscillatory (WENO), with a novelty consisting in the fact that nonlinear smoothness indicators are formulated in terms of the Gaussian likelihood of the local stencil data, replacing the conventional L2-type smoothness indicators of the original WENO method. We demonstrate that these GP-based smoothness indicators play a key role in the new algorithm, providing significant improvements in delivering high – and selectable – order accuracy in smooth flows, while successfully delivering non-oscillatory solution behavior in discontinuous flows.