We present a post-processing hybrid filter that is only applied to the approximation at the final time and allows for reducing errors away from a shock as well as near a shock for approximation with reduced stabilization applied during time-evolution. This filter is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a consistent data-driven Convolutional-Neural-Network (CNN) filter. While SIAC improves accuracy in smooth regions, it fails to reduce the O(1) errors near discontinuities, particularly in inviscid compressible flows with shocks. Our hybrid SIAC–CNN filter, trained exclusively on top-hat functions, enforces consistency constraints globally and higher-order moment conditions in smooth regions, reducing both ℓ2 and ℓ∞ errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate the effectiveness of the hybrid filter on the Euler equations for the Lax, Sod, and Shu–Osher shock-tube problems.

Contemporary evaluation and surgical approaches in congenital aortic valve disease have yielded limited success. The ability to evaluate and understand detailed flow characteristics surrounding surgical repair may be beneficial. This study explores the feasibility and utility of echocardiographic-based blood speckle imaging (BSI) in assessing pre- and post-operative flow characteristics in those with non-stenotic congenital aortic root disease undergoing aortic valve repair or valve-sparing root replacement (VSRR) surgery. Transesophageal echocardiogram was performed during the pre-operative and post-operative assessment surrounding aortic surgery for ten patients with non-stenotic congenital aortic root disease. BSI, utilizing block-matching algorithms, enabled detailed visualization and quantification of flow parameters from the echocardiographic data. Post-operative BSI unveiled enhanced hemodynamic patterns, characterized by quantified changes suggestive of the absence of stenosis and no more than trivial regurgitation. Rectification of an asymmetric jet and the reversal of flow on the posterior aspect of the ascending aorta resulted in a reduced oscillatory shear index ((Formula presented.)) of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.), increased peak wall shear stress of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.), and increased time-averaged wall shear stress of (Formula presented.) (pre-op) vs. (Formula presented.) (post-op) and (Formula presented.). This correction potentially attenuates cellular alterations within the endothelium. This study demonstrates that children and young adults with non-stenotic congenital aortic root disease undergoing valve-preserving operations experience significant improvements in flow dynamics within the left ventricular outflow tract and aortic root, accompanied by a reduction in (Formula presented.). These hemodynamic enhancements extend beyond the conventional echocardiographic assessments, offering immediate and valuable insights into the efficacy of surgical interventions.

The long-time evolution of extreme mass-ratio inspiral systems requires minimal phase and dispersion errors to accurately compute far-field waveforms, while high accuracy is essential near the smaller black hole (modeled as a Dirac delta distribution) for self-force computations. Spectrally accurate methods, such as nodal discontinuous Galerkin (DG) methods, are well suited for these tasks. Their numerical errors typically decrease as ∝(Δx)N+1, where Δx is the subdomain size and N is the polynomial degree of the approximation. However, certain DG schemes exhibit superconvergence, where truncation, phase, and dispersion errors can decrease as fast as ∝(Δx)2N+1. Superconvergent numerical solvers are, by construction, extremely efficient and accurate. We theoretically demonstrate that our DG scheme for the scalar Teukolsky equation with a distributional source is superconvergent, and this property is retained when combined with the hyperboloidal layer compactification technique. This ensures that waveforms, total energy and angular-momentum fluxes, and self-force computations benefit from superconvergence. We empirically verify this behavior across a family of hyperboloidal layer compactifications with varying degrees of smoothness. Additionally, we show that dissipative self-force quantities for circular orbits, computed at the point particle’s location, also exhibit a certain degree of superconvergence. Our results underscore the potential benefits of numerical superconvergence for efficient and accurate gravitational waveform simulations based on DG methods.

The simulation of plasma physics is computationally expensive because the underlying physical system is of high dimensions, requiring three spatial dimensions and three velocity dimensions. One popular numerical approach is Particle-In-Cell (PIC) methods owing to its ease of implementation and favorable scalability in high-dimensional problems. An unfortunate drawback of the method is the introduction of statistical noise resulting from the use of finitely many particles. In this paper we examine the application of the Smoothness-Increasing Accuracy-Conserving (SIAC) family of convolution kernel filters as denoisers for moment data arising from PIC simulations. We show that SIAC filtering is a promising tool to denoise PIC data in the physical space as well as capture the appropriate scales in the Fourier space. Furthermore, we demonstrate how the application of the SIAC technique reduces the amount of information necessary in the computation of quantities of interest in plasma physics such as the Bohm speed.

In this article we consider the extension of the (L)SIAC-MRA enhancement procedure to nonuniform meshes. We demonstrate that error reduction can be obtained on perturbed quadrilateral and Delaunay meshes, and investigate the effect of limited resolution and its impact on the procedure for various function types. We show that utilizing mesh -based localized kernel scalings, which were shown to reduce approximation errors for LSIAC filters, improve the performance of the LSIAC-MRA enhancement procedure. Lastly, we demonstrate the usefulness of enhanced approximations generated by (L)SIAC-MRA in mesh adaptivity applications, and show that SIAC reconstruction can be used in identification of regions of high error in steady-state DG approximations.

Filtering is a powerful tool in CFD that can aid in accurately and efficiently predicting the governing physics in simulations, leading to improved designs. Filters can remove subgrid scale high-frequency physics so that only large scale structures remain in the filtered solution, alleviate aliasing error, and mitigate Gibbs phenomenon. They can even extract hidden accuracy. The same ideas are useful in data compression, post-processing, and machine learning. Well-designed filters, such as the one that gives rise to the Smoothness-Increasing Accuracy-Conserving (SIAC) post-processing filters, can be used to extract hidden information in certain numerical simulations, creating even more accurate representations of the data. They can be adapted for boundaries, unstructured grids, and non-smooth solutions. Furthermore, well-designed filters have the potential to accurately capture multi-scale physics, and are flexible enough to combine simulation information with experimental data. The SIAC Magic Toolbox provides a codebase for efficient, effective, flexible filters for general data. It takes in two data files: one data file consisting of information on the mesh and a second data file consisting of information from the corresponding approximation, either modal or nodal data. If desired, the user can choose parameters that correlate to the amount of dissipation, accuracy, and scaling. Otherwise these parameters are set as default parameters. The toolbox then returns the filtered information in the same format.

This paper explores the discontinuous Galerkin (DG) methods for solving the Vlasov–Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG method provides an accurate numerical description with conservation and stability properties. This work studies the applicability of a post-processing technique to the DG solution in order to enhance its accuracy and resolution for the VM system. In particular, superconvergence in the negative-order norm for the probability distribution function and the electromagnetic fields is established for the DG solution. Numerical tests including Landau damping, two-stream instability, and streaming Weibel instabilities are considered showing the performance of the post-processor.

In this paper, we consider the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving partial differential equations (PDEs) with high order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy-Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. Previous theoretical results about this technique concentrated on first- and second-order equations. However, for linear higher order equations, Yan and Shu (J Sci Comput 17:27-47, 2002) numerically demonstrated that it is possible to improve the accuracy order to 2k + 1 for local discontinuous Galerkin (LDG) solutions using the SIAC filter. In this work, we firstly provide the theoretical proof for this observation. Furthermore, we prove the accuracy order of the ultra-weak local discontinuous Galerkin (UWLDG) solutions could be improved to 2k + 2 - m using the SIAC filter, where m = [n/2], n is the order of PDEs. Finally, we computationally demonstrate that for nonlinear higher order PDEs, we can also obtain a superconvergence approximation with the accuracy order of 2k + 1 or 2k + 2 - m by convolving the LDG solution and the UWLDG solution against the SIAC filter, respectively.

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.

In this article we introduce line Smoothness-Increasing Accuracy-Conserving Multi -Resolution Analysis. This is a procedure for exploiting convolution kernel post-processors for ob-taining more accurate multidimensional multiresolution analysis in terms of error reduction. This filtering-projection tool allows for the transition of data between different resolutions while simultane-ously decreasing errors in the fine grid approximation. It specifically allows for defining detail multi -wavelet coefficients when translating coarse data onto finer meshes. These coefficients are usually not defined in such cases. We show how to analytically evaluate the resulting convolutions and express the filtered approximation in a new basis. This is done by combining the filtering procedure with projection operators that allow for computational implementation of this scale transition procedure. Further, this procedure can be applied to piecewise constant approximations to functions, as it pro-vides error reduction. We demonstrate the effectiveness of this technique in two and three dimensions.