High-order Accuracy In Smooth Regions Research Articles

Approximation of piecewise smooth functions by nonlinear bivariate C2 quartic spline quasi-interpolants on criss-cross triangulations

In this paper we focus on the space of C2 quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Nonlinear 2D C1 Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

A new smoothed particle hydrodynamics method based on high-order moving-least-square targeted essentially non-oscillatory scheme for compressible flows

In this study, we establish a hybrid high-order smoothed particle hydrodynamics (SPH) framework (MLS-TENO-SPH) for compressible flows with discontinuities, which is able to achieve genuine high-order convergence in smooth regions and also capture discontinuities well in non-smooth regions. The framework can be either fully Lagrangian, Eulerian or realizing arbitary-Lagrangian-Eulerian (ALE) feature enforcing the isotropic particle distribution in specific cases. In the proposed framework, the computational domain is divided into smooth regions and non-smooth regions, and these two regions are determined by a strong scale separation strategy in the targeted essentially non-oscillatory (TENO) scheme. In smooth regions, the moving-least-square (MLS) approximation is used for evaluating high-order derivative operator, which is able to realize genuine high-order construction; in non-smooth regions, the new TENO scheme based on Vila's framework with several new improvements will be deployed to capture discontinuities and high-wavenumber flow scales with low numerical dissipation. The present MLS-TENO-SPH method is validated with a set of challenging cases based on the Eulerian, Lagrangian or ALE framework. Numerical results demonstrate that the MLS-TENO-SPH method features lower numerical dissipation and higher efficiency than the conventional method, and can restore genuine high-order accuracy in smooth regions. Overall, the proposed framework serves as a new exploration in high-order SPH methods, which are potential for compressible flow simulations with shockwaves.

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

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

An improved simple WENO limiter for discontinuous Galerkin methods solving hyperbolic systems on unstructured meshes

In this paper, we improve the simple WENO limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method in [34] for solving two-dimensional hyperbolic systems on unstructured meshes. The major improvement is reducing the number of polynomials transformed to the characteristic fields for each direction. For the triangular mesh, this new simple WENO limiter transforms only two polynomials to the characteristic fields for each direction, while the original simple WENO limiter [34] uses four polynomials. Thus the improved simple WENO limiter reduces the computational cost and improves the efficiency. It provides a simpler and more practical and efficient way to the characteristic-wise limiting procedure, while still simultaneously maintaining uniform high-order accuracy in smooth regions and controlling spurious nonphysical oscillations near discontinuities. We also apply this improved simple WENO limiter to a high order method constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds [10], where polygonal meshes are constructed based on random points and the traditional DG method was adopted on the constructed polygonal mesh. For such a complex polygonal mesh, this limiter still transforms only two polynomials to the characteristic fields. Thus the simplicity and efficiency of this new limiter is more evident in this case. Numerical results are provided to illustrate the accuracy and effectiveness of this procedure. For some examples, the improved limiter has less smearing and higher resolution than the original one.

A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a shock-capturing Finite Volume (FV) scheme coupled via an hp adaptive mesh refinement (hp-AMR) strategy that offers high-order accurate resolution of the embedded geometries. The hp-AMR strategy is based on a multi-level block-structured domain partition in which each level is represented by block-structured Cartesian grids and the embedded geometry is represented implicitly by a level set function. The intersection of the embedded geometry with the grids produces the implicitly-defined mesh that consists of a collection of regular rectangular cells plus a relatively small number of irregular curved elements in the vicinity of the embedded boundaries. High-order quadrature rules for implicitly-defined domains enable high-order accuracy resolution of the curved elements with a cell-merging strategy to address the small-cell problem. The hp-AMR algorithm treats the system with a second-order finite volume scheme at the finest level to dynamically track the evolution of solution discontinuities while using dG schemes at coarser levels to provide high-order accuracy in smooth regions of the flow. On the dG levels, the methodology supports different orders of basis functions on different levels. The space-discretized governing equations are then advanced explicitly in time using high-order Runge-Kutta algorithms. Numerical tests are presented for two-dimensional and three-dimensional problems involving an ideal gas. The results are compared with both analytical solutions and experimental observations and demonstrate that the framework provides high-order accuracy for smooth flows and accurately captures solution discontinuities.

A conservative discontinuous Galerkin discretization for the chemically reacting Navier-Stokes equations

We present a detailed description and verification of a discontinuous Galerkin finite element method (DG) for the multi-component chemically reacting compressible Navier-Stokes equations that retains the desirable properties of DG, namely discrete conservation and high-order accuracy in smooth regions of the flow. Pressure equilibrium between adjacent elements is maintained through the consistent evaluation of the thermodynamics model and the resulting weak form, as well as the proper choice of nodal basis. As such, the discretization does not generate unphysical pressure oscillations in smooth regions of the flow or at material interfaces where the temperature is continuous. Additionally, we present an hp-adaptive DG method for solving systems of ordinary differential equations, DGODE, which is used to resolve the temporal evolution of the species concentrations due to stiff chemical reactions. The coupled solver is applied to several challenging test problems including multi-component shocked flows as well as chemically reacting detonations, deflagrations, and shear flows with detailed kinetics. We demonstrate that the discretization does not produce unphysical pressure oscillations and, when applicable, we verify that it maintains discrete conservation. The solver is also shown to reproduce the expected temperature and species profiles throughout a detonation as well as the expected two-dimensional cellular detonation structure. We also demonstrate that the solver can produce accurate, high-order, approximations of temperature and species profiles without artificial stabilization for the case of a one-dimensional pre-mixed flame. Finally, high-order solutions of two- and three-dimensional multi-component chemically reacting shear flows, computed without any additional stabilization, are presented.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

New Finite Difference Hermite WENO Schemes for Hamilton–Jacobi Equations

In this paper, new finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving the Hamilton–Jacobi equations on structured meshes. The crucial idea of the spatial reconstructions is borrowed from the original HWENO schemes (Qiu and Shu in J Comput Phys 204:82–99, 2005), in which the function and its first derivative values are evolved in time and used in the reconstruction. Such new HWENO spatial reconstructions with the application of three unequal-sized spatial stencils result in an important innovation that we perform only spatial HWENO reconstructions for numerical fluxes of function values and high-order linear reconstructions for numerical fluxes of derivatives, which are different to other HWENO schemes. The new HWENO schemes could obtain smaller errors with optimal high-order accuracy in smooth regions, and keep sharp transitions and non-oscillatory property near discontinuities. Extensive benchmark examples are performed to illustrate the good performance of such new finite difference HWENO schemes.

A novel fourth-order WENO interpolation technique

Context. Several numerical problems require the interpolation of discrete data that present at the same time (i) complex smooth structures and (ii) various types of discontinuities. The radiative transfer in solar and stellar atmospheres is a typical example of such a problem. This calls for high-order well-behaved techniques that are able to interpolate both smooth and discontinuous data. Aims. This article expands on different nonlinear interpolation techniques capable of guaranteeing high-order accuracy and handling discontinuities in an accurate and non-oscillatory fashion. The final aim is to propose new techniques which could be suitable for applications in the context of numerical radiative transfer. Methods. We have proposed and tested two different techniques. Essentially non-oscillatory (ENO) techniques generate several candidate interpolations based on different substencils. The smoothest candidate interpolation is determined from a measure for the local smoothness, thereby enabling the essentially non-oscillatory property. Weighted ENO (WENO) techniques use a convex combination of all candidate substencils to obtain high-order accuracy in smooth regions while keeping the essentially non-oscillatory property. In particular, we have outlined and tested a novel well-performing fourth-order WENO interpolation technique for both uniform and nonuniform grids. Results. Numerical tests prove that the fourth-order WENO interpolation guarantees fourth-order accuracy in smooth regions of the interpolated functions. In the presence of discontinuities, the fourth-order WENO interpolation enables the non-oscillatory property, avoiding oscillations. Unlike Bézier and monotonic high-order Hermite interpolations, it does not degenerate to a linear interpolation near smooth extrema of the interpolated function. Conclusion. The novel fourth-order WENO interpolation guarantees high accuracy in smooth regions, while effectively handling discontinuities. This interpolation technique might be particularly suitable for several problems, including a number of radiative transfer applications such as multidimensional problems, multigrid methods, and formal solutions.

Development of a mass-preserving level set redistancing algorithm for simulation of rising bubble

This article is aimed to simulate the gas-liquid flow of rising bubbles with a mass-preserving level set method. To resolve the topological changes of gas-liquid interface where the classic finite difference scheme may yield oscillation solutions, the spatial terms in the level set advection equation will be approximated by an optimized compact reconstruction weighted essentially non-oscillatory (OCRWENO) scheme. This scheme achieves high-order accuracy in smooth regions, and meanwhile avoid numerical oscillation near discontinuities. Two benchmark problems including vortex flow and deforming field are chosen to compare the present simulation with previous numerical researches. Several rising bubble problems are validated by the proposed level set method.

SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics

We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads.

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

This paper establishes the entropy convergence of a new two-value high resolution finite volume scheme with slope relaxation for conservation laws. This scheme, motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities, unifies and evolves slopes directly with a slope relaxation equation that governs the evolution of slopes in both smooth and discontinuous regions. Proper choices of slopes are realized adaptively via a relaxation parameter. The scheme is shown to be total-variation-bounded (TVB) stable and satisfies cell-entropy inequalities.

A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

Although the Discontinuous Galerkin (dg) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, high-order accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie–Grüneisen family. Within the dg framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one- and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (gpu) to achieve high speedup.

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper, we use Daubechies scaling functions as test functions for the Galerkin method, and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations. It can be proved that the schemes are TVD schemes. Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations. Furthermore, they have high-order accuracy in smooth regions and good resolution of singularities.

High-Order WENO Schemes for Hamilton--Jacobi Equations on Triangular Meshes

In this paper we construct high-order weighted essentially nonoscillatory (WENO) schemes for solving the nonlinear Hamilton--Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Both third-order and fourth-order WENO schemes using combinations of second-order approximations with nonlinear weights are constructed. Extensive numerical experiments are performed to demonstrate the stability and accuracy of the methods. High-order accuracy in smooth regions, good resolution of derivative singularities, and convergence to viscosity solutions are observed.

High-Order Monotonicity-Preserving Compact Schemes for Linear Scalar Advection on 2-D Irregular Meshes

This paper is concerned with the numerical solution for linear scalar advection problems, the velocity field of which may be uniform or a given function of the space variable. We would like to propose the following: (1) a new family of 1-D compact explicit schemes, which preserve monotonicity while maintaining high-order accuracy in smooth regions; and (2) an extension to the 2-D case of this family of schemes, which ensures good accuracy and isotropy of the computed solution even for very distorted meshes. A few theoretical results are proven, while abundant numerical tests are shown in order to illustrate the quality of the schemes at issue.

A Finite-Volume High-Order ENO Scheme for Two-Dimensional Hyperbolic Systems

We consider here the finite-volume approach in developing a two-dimensional, high-order accurate, essentially non-oscillatory shock-capturing scheme. Such a scheme achieves high-order spatial accuracy by a piecewise polynomial approximation of the solution from its cell averages. High-order Runge-Kutta methods are employed for time integration, thus making such schemes best-suited for unsteady problems. The focal point in our development is a high-order spatial operator which will retain high-order accuracy in smooth regions, yet avoid the oscillatory behavior that is associated with interpolation across steep gradients. Such an operator is first presented within the context of a scalar function on a rectangular mesh and then extended to hyperbolic systems of equations and curvilinear meshes. Spatial and temporal accuracy are validated through grid refinement studies, involving the solutions of scalar hyperbolic equations and the Euler equations of gas dynamics. Through a control-volume approach, we find that this two-dimensional scheme is readily applied to inviscid flow problems involving solid walls and non-trivial geometries. Results of a physically relevant, numerical experiment are presented for qualitative and quantitative examination.

High-Order Essentially Nonoscillatory Schemes for Hamilton–Jacobi Equations

Hamilton–Jacobi (H–J) equations are frequently encountered in applications, e.g., in control theory and differential games. H–J equations are closely related to hyperbolic conservation laws—in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H–J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.