Smoothness-Increasing Accuracy-Conserving Research Articles

How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Higher order accuracy is one of the well-known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. The smoothness-increasing accuracy-conserving (SIAC) filter is a popular post-processing technique introduced by Cockburn et al. (Math. Comput. 72(242): 577–606, 2003). It can raise the convergence rate of the DG solution (with a polynomial of degree k) from order \(k+1\) to order \(2k+1\) in the \(L^2\) norm. This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction. The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features, such as extra smoothness. Second, we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We prove the superconvergence error estimate of the new SIAC filters. Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.

Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods

This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifically, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) filtering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of different resolutions. Although this article presents the details of the SIAC filter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.

Residual Estimates for Post-processors in Elliptic Problems

In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that “tweaks” a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Conserving filter and superconvergent patch recovery. Extensive numerical tests are conducted that confirm our analytic findings.

Enhancing accuracy with a convolution filter: What works and why!

In this paper we present a simplified discussion of the Smoothness-Increasing Accuracy-Conserving (SIAC) filter. We demonstrate the importance of appropriately initializing the data in order to be able to extract higher orders of accuracy by comparing the nodal and modal forms of a discontinuous Galerkin approximation applied to a simplified cubic polynomial. Using the modal form, we are able to exactly reproduce the cubic polynomial, whereas this reconstruction does not occur using the nodal form. Furthermore, we tie the ability of the filter to extract extra accuracy to its accurate wave propagation properties.

Superconvergence and the Numerical Flux: a Study Using the Upwind-Biased Flux in Discontinuous Galerkin Methods

One of the beneficial properties of the discontinuous Galerkin method is the accurate wave propagation properties. That is, the semi-discrete error has dissipation errors of order 2k+1 (le Ch^{2k+1}) and order 2k+2 for dispersion (le Ch^{2k+2}). Previous studies have concentrated on the order of accuracy, and neglected the important role that the error constant, C, plays in these estimates. In this article, we show the important role of the error constant in the dispersion and dissipation error for discontinuous Galerkin approximation of polynomial degree k, where k=0,1,2,3. This gives insight into why one may want a more centred flux for a piecewise constant or quadratic approximation than for a piecewise linear or cubic approximation. We provide an explicit formula for these error constants. This is illustrated through one particular flux, the upwind-biased flux introduced by Meng et al., as it is a convex combination of the upwind and downwind fluxes. The studies of wave propagation are typically done through a Fourier ansatz. This higher order Fourier information can be extracted using the smoothness-increasing accuracy-conserving (SIAC) filter. The SIAC filter ties the higher order Fourier information to the negative-order norm in physical space. We show that both the proofs of the ability of the SIAC filter to extract extra accuracy and numerical results are unaffected by the choice of flux.

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (J Sci Comput 77:579–596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics equations to several standard test problems with a variety of boundary conditions.

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions over Nonuniform Meshes: Superconvergence and Optimal Accuracy

Smoothness-increasing accuracy-conserving (SIAC) filtering is an area of increasing interest because it can extract the “hidden accuracy” in discontinuous Galerkin (DG) solutions. It has been shown that by applying a SIAC filter to a DG solution, the accuracy order of the DG solution improves from order k+1 to order 2k+1 for linear hyperbolic equations over uniform meshes. However, applying a SIAC filter over nonuniform meshes is difficult, and the quality of filtered solutions is usually unsatisfactory applied to approximations defined on nonuniform meshes. The applicability to such approximations over nonuniform meshes is the biggest obstacle to the development of a SIAC filter. The purpose of this paper is twofold: to study the connection between the error of the filtered solution and the nonuniform mesh and to develop a filter scaling that approximates the optimal error reduction. First, through analyzing the error estimates for SIAC filtering, we computationally establish for the first time a relation between the filtered solutions and the unstructuredness of nonuniform meshes. Further, we demonstrate that there exists an optimal accuracy of the filtered solution for a given nonuniform mesh and that it is possible to obtain this optimal accuracy by the method we propose, an optimal filter scaling. By applying the newly designed filter scaling over nonuniform meshes, the filtered solution has demonstrated improvement in accuracy order as well as reducing the error compared to the original DG solution. Finally, we apply the proposed methods over a large number of nonuniform meshes and compare the performance with existing methods to demonstrate the superiority of our method.

Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrödinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrödinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrödinger equations by using alternating numerical fluxes.

Multi-Dimensional Filtering: Reducing the Dimension Through Rotation

Over the past few decades there has been a strong effort toward the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this postprocessor. These advantages can be exploited during flow visualization, for example, by applying the SIAC filter to DG data before streamline computations [M. Steffen, S. Curtis, R. M. Kirby, and J. K. Ryan, IEEE Trans. Vis. Comput. Graphics, 14 (2008), pp. 680--692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [D. Walfisch, J. K. Ryan, R. M. Kirby, and R. Haimes, J. Sci. Comput., 38 (2009), pp. 164--184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure, and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha -th order (1 le alpha le {k+1}) divided difference of the DG error in the L^2 norm is of order {k + frac{3}{2} - frac{alpha }{2}} when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order {2k + frac{3}{2} - frac{alpha }{2}} in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least ({frac{3}{2}k+1})th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order {k+1} in the L^2 norm for the divided differences of DG errors and thus ({2k+1})th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

Nonuniform Discontinuous Galerkin Filters via Shift and Scale

Convolving the output of discontinuous Galerkin computations with symmetric smoothness-increasing accuracy-conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. This paper interprets these filters as instances of a general class of position-dependent (PSIAC) spline filters that can have nonuniform knot sequences and skip B-splines of the sequence. PSIAC filters with rational knot sequences have rational coefficients. For prototype knot sequences, such as integer sequences that may have repeated entries, PSIAC filters can be expressed in symbolic form. Based on the insight that filters for shifted or scaled knot sequences are easily derived by nonuniform scaling of one prototype filter, a single filter can be reused in different locations and at different scales. Computing a value of the convolution then simplifies to forming a scalar product of a short vector with the local output data. Restating one-sided filters in this form improves both stability and efficiency compared to their original formulation via numerical integration. PSIAC filtering is demonstrated for several established and one new boundary filter.

Smoothness-Increasing Accuracy-Conserving (SIAC) filters for derivative approximations of discontinuous Galerkin (DG) solutions over nonuniform meshes and near boundaries

Accurate approximations for the derivatives are usually required in many application areas such as biomechanics, chemistry and visualization applications. With the help of Smoothness-Increasing Accuracy-Conserving (SIAC) filtering, one can enhance the derivatives of a discontinuous Galerkin solution. However, current investigations of derivative filtering are limited to uniform meshes and periodic boundary conditions, which do not meet practical requirements. The purpose of this paper is twofold: to extend derivative filtering to nonuniform meshes and propose position-dependent derivative filters to handle filtering near the boundaries. Through analyzing the error estimates for SIAC filtering, we extend derivative filtering to nonuniform meshes by changing the scaling of the filter. For filtering near boundaries, we discuss the advantages and disadvantages of two existing position-dependent filters and then extend them to position-dependent derivative filters, respectively. Further, we prove that with the position-dependent derivative filters, the filtered solutions can obtain a better accuracy rate compared to the original discontinuous Galerkin approximation with arbitrary derivative orders over nonuniform meshes. One- and two-dimensional numerical results are provided to support the theoretical results and demonstrate that the position-dependent derivative filters, in general, enhance the accuracy of the solution for both uniform and nonuniform meshes.

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called smoothness-increasing accuracy-conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties.

General spline filters for discontinuous Galerkin solutions

The discontinuous Galerkin (dG) method outputs a sequence of polynomial pieces. Post-processing the sequence by Smoothness-Increasing Accuracy-Conserving (SIAC) convolution not only increases the smoothness of the sequence but can also improve its accuracy and yield superconvergence. SIAC convolution is considered optimal if the SIAC kernels, in the form of a linear combination of B-splines of degree d, reproduce polynomials of degree 2d. This paper derives simple formulas for computing the optimal SIAC spline coefficients for the general case including non-uniform knots.

One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-uniform Meshes

In this paper, we introduce a new position-dependent smoothness-increasing accuracy-conserving (SIAC) filter that retains the benefits of position dependence as proposed in van Slingerland et al. (SIAM J Sci Comput 33:802–825, 2011) while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of \(4k+1\) central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of \(k\ge 3\) for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues—maintaining the same support size and with similar function evaluation characteristics as the symmetric filter in a way that has better numerical conditioning—making it, unlike its predecessor, amenable for GPU computing. Our new filter was conceived by revisiting the original error analysis for superconvergence of SIAC filters and by examining the role of the B-splines and their weights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that our new filter is globally superconvergent for \(k=1\) and superconvergent in the interior (e.g., region excluding the boundary) for \(k\ge 2\). Furthermore, we present the first theoretical proof of superconvergence for postprocessing over smoothly varying meshes, and explain the accuracy-order conserving nature of this new filter when applied to certain non-uniform meshes cases. We provide numerical examples supporting our theoretical results and demonstrating that our new filter, in general, enhances the smoothness and accuracy of the solution. Numerical results are presented for solutions of both linear and nonlinear equations solved on both uniform and non-uniform one- and two-dimensional meshes.

Superconvergent error estimates for position-dependent smoothness-increasing accuracy-conserving (SIAC) post-processing of discontinuous Galerkin solutions

Superconvergence of discontinuous Galerkin methods is an area of increasing interest due to the ease with which higher order information can be extracted from the approximation. Cockburn, Luskin, Shu, and Suli showed that by applying a B-spline filter to the approximation at the final time, the order of accuracy can be improved from order k+1 to order 2k+1 in the L2-norm for linear hyperbolic equations with periodic boundary conditions (where k is the polynomial degree and h is the mesh element diameter) [Math. Comp. (2003)]. The applicability of this filter for linear hyperbolic problems with non-periodic boundary conditions was computationally extended and renamed a position-dependent smoothness-increasing accuracy-conserving (SIAC) filter by van Slingerland, Ryan, Vuik [SISC (2011)]. However, error estimates in the L2$-norm for this new position-dependent SIAC filter were never given. Furthermore, error estimates in the L-infinity-norm have not been established for the original kernel nor the position-dependent kernel. In this paper, for the first time we establish that it is possible to obtain order s, s=min{2k+1,2k + 2-\frac {d}{2}} accuracy in the L-infinity-norm for the position-dependent SIAC filter, where d is the dimension. Furthermore, we extend the error estimates given by Cockburn et al. so that they are applicable to the entire domain when implementing the position-dependent SIAC filter. We also computationally demonstrate the applicability of this filter for visualization of streamlines.

Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions: Application to Structured Tetrahedral Meshes

In this paper, we attempt to address the potential usefulness of smoothness-increasing accuracy-conserving (SIAC) filters when applied to real-world simulations. SIAC filters as a class of post-processors were initially developed in Bramble and Schatz (Math Comput 31:94, 1977) and later applied to discontinuous Galerkin (DG) solutions of linear hyperbolic partial differential equations by Cockburn et al. (Math Comput 72:577, 2003), and are successful in raising the order of accuracy from $$k+1$$ k + 1 to $$2k+1$$ 2 k + 1 in the $$L^2$$ L 2 --norm when applied to a locally translation-invariant mesh. While there have been several attempts to demonstrate the usefulness of this filtering technique to nontrivial mesh structures (Curtis et al. in SIAM J Sci Comput 30(1):272, 2007; Mirzaee et al. in SIAM J Numer Anal 49:1899, 2011; King et al. in J Sci Comput, 2012), the application of the SIAC filter never exceeded beyond two-space dimensions. As tetrahedral meshes are often the type considered in more realistic simulations, we contribute to the class of SIAC post-processors by demonstrating the effectiveness of SIAC filtering when applied to structured tetrahedral meshes. These types of meshes are generated by tetrahedralizing uniform hexahedra and therefore, while maintaining the structured nature of a hexahedral mesh, they exhibit an unstructured tessellation within each hexahedral element. Moreover, we address the computationally intensive task of performing numerical integrations when one considers tetrahedral elements for SIAC filtering and provide guidelines on how to ameliorate these challenges through the use of more general cubature rules. We consider two examples of a hyperbolic equation and confirm the usefulness of SIAC filters in obtaining the superconvergence accuracy of $$2k+1$$ 2 k + 1 when applied to structured tetrahedral meshes. Additionally, the DG methodology merely requires weak constraints on the fluxes between elements. As SIAC filters improve this weak continuity to $$\mathcal{C }^{k-1}$$ C k - 1 --continuity at the element interfaces, we provide results that show how post-processing is useful in extracting smooth isosurfaces of DG fields.

Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Solutions over Unstructured Triangular Meshes

The discontinuous Galerkin (DG) method has very quickly found utility in such diverse applications as computational solid mechanics, fluid mechanics, acoustics, and electromagnetics. The DG methodology merely requires weak constraints on the fluxes between elements. This feature provides a flexibility which is difficult to match with conventional continuous Galerkin methods. However, allowing discontinuity between element interfaces can in turn be problematic during simulation postprocessing, such as in visualization. Consequently, smoothness-increasing accuracy-conserving (SIAC) filters were proposed in [M. Steffen et al., IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 680--692, D. Walfisch et al., J. Sci. Comput., 38 (2009), pp. 164--184] as a means of introducing continuity at element interfaces while maintaining the order of accuracy of the original input DG solution. Although the DG methodology can be applied to arbitrary triangulations, the typical application of SIAC filters has been to DG solutions obtained over translation invariant meshes such as structured quadrilaterals and triangles. As the assumption of any sort of regularity including the translation invariance of the mesh is a hindrance towards making the SIAC filter applicable to real life simulations, we demonstrate in this paper for the first time the behavior and complexity of the computational extension of this filtering technique to fully unstructured tessellations. We consider different types of unstructured triangulations and show that it is indeed possible to get reduced errors and improved smoothness through a proper choice of kernel scaling. These results are promising, as they pave the way towards a more generalized SIAC filtering technique.

Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws

In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this "hidden accuracy" through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to $$(2k+m)$$ -th order in the negative-order norm, where $$m$$ depends upon the chosen flux.

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy

Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order \(k+\frac{1}{2}\) to order 2k+1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577–606, 2003; Curtis et al. in SIAM J. Sci. Comput. 30(1):272–289, 2007; Mirzaee et al. in SIAM J. Numer. Anal. 49:1899–1920, 2011). Additionally, it improves the weak continuity in the discontinuous Galerkin method to k−1 continuity. Typically this improvement has a positive impact on the error quantity in the sense that it also reduces the absolute errors. However, not enough emphasis has been placed on the difference between superconvergent accuracy and improved errors. This distinction is particularly important when it comes to understanding the interplay introduced through meshing, between geometry and filtering. The underlying mesh over which the DG solution is built is important because the tool used in SIAC filtering—convolution—is scaled by the geometric mesh size. This heavily contributes to the effectiveness of the post-processor. In this paper, we present a study of this mesh scaling and how it factors into the theoretical errors. To accomplish the large volume of post-processing necessary for this study, commodity streaming multiprocessors were used; we demonstrate for structured meshes up to a 50× speed up in the computational time over traditional CPU implementations of the SIAC filter.