Abel Transform Research Articles

On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws

We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this context. To do so, we introduce discontinuous-Galerkin-like interface terms for multi-block upwind SBP methods applied to nonlinear conservation laws. We investigate the behavior of the upwind SBP methods for flux vector splittings of varying complexity on Cartesian as well as unstructured curvilinear multi-block meshes. Moreover, we analyze the local linear/energy stability of these methods following Gassner, Svärd, and Hindenlang (2022, doi:10.1007/s10915-021-01720-8). Finally, we investigate the robustness of upwind SBP methods for challenging examples of shock-free flows of the compressible Euler equations such as a Kelvin-Helmholtz instability and the inviscid Taylor-Green vortex.

Efficient entropy-stable discontinuous spectral-element methods using tensor-product summation-by-parts operators on triangles and tetrahedra

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-conservative modal formulation, which is rendered entropy stable when a dissipative numerical flux is used at element interfaces. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol–Koornwinder–Dubiner polynomial basis used to represent the numerical solution on the reference triangle or tetrahedron, and a weight-adjusted approximation is employed in order to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, we obtain an improvement in time complexity from O(p2d) to O(pd+1) relative to existing entropy-stable formulations using multidimensional SBP operators not possessing such a tensor-product structure, where p is the polynomial degree of the approximation and d is the number of spatial dimensions. The number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is accordingly reduced by a factor ranging from 1.56 at p=2 to 4.57 at p=10 for triangles, and from 1.88 at p=2 to 10.99 at p=10 for tetrahedra. Through numerical experiments involving smooth solutions to the compressible Euler equations on isoparametric triangular and tetrahedral grids, the proposed methods using tensor-product SBP operators are shown to exhibit similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional operators based on symmetric quadrature rules, with both approaches achieving order p+1 convergence with respect to the element size in the presence of interface dissipation as well as exponential convergence with respect to the polynomial degree. Furthermore, both operator families are shown to give rise to entropy-stable schemes which exhibit excellent robustness for test problems characteristic of under-resolved turbulence simulations. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.

An optimized design of delay-and energy-efficient Booth multiplier

Multipliers are essential computation units in virtually all computing systems, including processors and numerous AI accelerator architectures. This paper presents an optimized architecture for a Booth multiplier, targeting high performance while minimizing energy consumption and area utilization. The design optimization focuses on all three multiplier stages: partial product generation, reduction, and summation. To enhance delay and energy efficiency in the partial product generation stage, we first employed a simplified configuration comprising inverters and a sign selection unit instead of complex binary-to-two's complement circuitry. Next, to achieve further delay and area efficiency at this stage, logic optimization is applied at the partial product's generation circuitry by designing Booth encoders to remove redundant logic in multiplexers circuitry. Moreover, we introduced specialized sign compressors tailored for carry-save compression in the compression stage. Compared to conventional counterparts, these compressors offered lower power consumption and reduced critical path delay with only two XOR logic gates. Finally, in the summation stage, we proposed an optimized design segment for Carry Look-Ahead Adder for the final summation stage, designed to deliver swift throughput with minimal fan-in logic gates, even in the context of high bit-width configurations. This segment is cascaded to make a 13-bit final adder for the summation stage in the proposed design. The proposed architecture undergoes ASIC-targeted synthesis in Cadence Genus employing FreePDK CMOS 45 nm process technology. Synthesized results, along with theoretical design complexity comparison, demonstrate that the proposed design surpasses state-of-the-art 8 × 8 multiplier designs by critical metrics, including delay, power consumption, area utilization, power delay product, and area delay product.

Spline-based Abel Transform (SAT) radial property reconstruction for noise and trapping correction: Application to axisymmetric sooting flames

In combustion research, optical diagnostics often necessitate Abel inversion to deconvolve the measurements of axisymmetric flames, especially to determine local soot temperature fields from integrated flame radiative emission measurements. However, traditional Abel inversion methods can considerably amplify experimental noise, especially towards the flame centerline. Although various strategies exist to mitigate this issue, they typically do not address the correction of signal trapping within the flame. The present paper introduces an adapted Abel inversion tool that combines resistance to experimental noise with a mechanism to correct signal trapping effects. The tool employs noise-free Abel inversion using piecewise spline functions. Its effectiveness is validated through numerical comparisons with conventional methods. The practical application is demonstrated in the measurement of soot temperature in a canonical laminar diffusion ethylene flame. This is achieved by using multi-wavelength line-of-sight attenuation (LOSA) and emission measurements. The tool corrects for the signal trapping effect in emission data by integrating extinction profiles, thus enabling a more precise soot temperature analysis. The current study also compares two techniques of temperature measurement based on thermal emission: traditional two-color pyrometry and a calibrated emission and absorption ratio method. The study further explores the effects of signal trapping, scattering contribution, and the absorption function ratio on soot temperature determination based on experimental profiles. The findings suggest that signal trapping and scattering contributions have a negligible impact on measurements at the wavelengths examined. However, the choice of absorption function ratio significantly influences the outcomes of two-color pyrometry, highlighting the advantage of the one color emission/absorption technique for the determination of sooting flames temperature measurements.

A laser-induced plasma analysis based on the inversion of Abel transformation

A laser-induced plasma analysis based on the inversion of Abel transformation

General q-series transformations based on Abel's lemma on summation by parts and their applications

In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among include some new quadratic, cubic, and quartic transformations. Furthermore, we put forward the so-called ( R , S ) -type transformation with arbitrary degree to unify such multibasic transformations. A special ( 3 , 3 ) -type transformation and its example are presented.

An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form

<p>We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.</p>

One‐Dimensional Variational Ionospheric Retrieval Using Radio Occultation Bending Angles: 2. Validation

AbstractCulverwell et al. (2023, https://doi.org/10.1029/2023SW003572) described a new one‐dimensional variational (1D‐Var) retrieval approach for ionospheric GNSS radio occultation (GNSS‐RO) measurements. The approach maps a one‐dimensional ionospheric electron density profile, modeled with multiple “Vary‐Chap” layers, to bending angle space. This paper improves the computational performance of the 1D‐Var retrieval using an improved background model and validates the approach by comparing with the COSMIC‐2 profile retrievals, based on an Abel Transform inversion, and co‐located (within 200 km) ionosonde observations using all suitable data from 2020. A three or four layer Vary‐Chap in the 1D‐Var retrieval shows improved performance compared to COSMIC‐2 retrievals in terms of percentage error for the F2 peak parameters (NmF2 and hmF2). Furthermore, skill in retrieval (compared to COSMIC‐2 profiles) throughout the bottomside (∼90–300 km) has been demonstrated. With a single Vary‐Chap layer the performance is similar, but this improves by approximately 40% when using four‐layers.

A study of Griffith crack in nonlocal infinite magneto-elastic media

The main objective of this article is to study how a finite crack interacts with horizontally polarized shear (SH) waves in a nonlocal isotropic medium when a magnetic field is present. The study aims to understand the effects of various parameters, including the magnetic field, non-locality, frequency of SH waves, on the behavior of the crack. Here, we are concerned with Griffith’s theory of a brittle fracture crack mechanism. The article starts by formulating the governing equations for the propagation of SH waves in the given medium. These equations are converted into dual integral equations using Fourier transformation. Subsequently, an Abel transformation is applied to obtain a Fredholm integral equation of the second kind. Daubechies wavelet family has been used to find an approximate solution for the integral equation. The article aims to determine specific physical quantities including stress intensity factor (SIF), shear stress, and crack energy related to the behavior of the crack. These parameters are essential in understanding how the crack responds to the external conditions. The application of Daubechies wavelets in this context is a relatively new and innovative approach. The study performed a comprehensive analysis of the effects of various factors on the crack behavior. The findings are presented in graphical form to help illustrate the conclusions. To assess the accuracy of the proposed method, the obtained results are compared with known results from the literature for local media. This is an important step in demonstrating the reliability and effectiveness of the proposed approach.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

A stable scheme of the Curvilinear Shallow Water Equations with no-penetration and far-field boundary conditions

This paper presents a stable and highly accurate numerical tool for computing river flows in urban areas, which is a first step towards a numerical tool for flood predictions. We start with the (linearized) well-posedness analysis by Ghader and Nordström (2014), where far-field boundary conditions were proposed and extend their analysis to include wall boundaries. Specifically, we employed high-order Summation-by-parts (SBP) finite-difference operators to construct a scheme for the Shallow Water Equations. We also developed a stable SBP scheme with Simultaneous Approximation Terms that impose far-field and wall boundaries. Finally, we extended the schemes and their stability proofs to non-Cartesian domains. To demonstrate the strength of the schemes, we performed computations for problems with exact solutions to obtain second, third, and fourth (2, 3, 4) convergence rates. Finally, we applied the 4th-order scheme to steady river channels, the canal (or flood-control channel simulations), and dam-break problems. The results show that the imposition of the boundary conditions is stable, and the far-field boundaries cause no visible reflections at the boundaries.

Modeling the impact of sulfide ore processing waste disposal facilities on the environment and human health

After the completion of the exploitation of ore deposits, atmospheric migration of sulfur compounds remains an urgent ecological and hygienic problem. The aim of the study was to obtain scientifically sound ideas about the danger of waste from the extraction and processing of sulfide ores of non-ferrous and precious metals accumulated in mining regions for the environment and public health based on the methodology of integrated modeling. The emission of sulfur compounds was studied on a full-scale model of the Ursky dump of barite–pyrite bulk (Ursk settlement, Kemerovo Region), the nature of their combined action was studied in a model experiment on Wistar rats. According to the simulation results, the main components of harmful emissions from the Ursky dump include dimethyl sulfide, dimethyl sulfoxide and sulfur dioxide, the concentrations of which in the air exceeded the normative ones by 129–170, 119–127 and 1,7 times. When exposing experimental animals with a two-component mixture of dimethyl sulfide and dimethyl sulfoxide, a partial summation of the neuro- and hepatotoxic effects of the components was noted, while a three-component mixture of dimethyl sulfide, dimethyl sulfoxide and sulfur dioxide potentiated the irritating effect. The dependences of the qualitative and quantitative characteristics of the biological effects of the isolated mixtures on the sum of the concentrations of their components and time are investigated. The proven algorithm for modeling the effects of the impact of a sulfide-containing dump on the environment and public health makes it possible to expand the possibilities of environmental expertise of environmental protection measures and assessment of risks to public health in mining regions.

Generalization to a wider class of entropy split methods for compressible ideal MHD

The high order entropy split methods of Sjögreen & Yee (2019, 2021), by entropy splitting of the compressible Euler (inviscid) flux derivatives for a thermally-perfect gas are based on Harten’s entropy function (Harten, 1983; Gerritsen and Olsson, 1996; Yee et al., 2000). Their formulation have been proven entropy conserving and stable by taking advantage of the homogeneity property of Euler flux, symmetrizable Euler flux derivatives and energy-norm stability in conjunction with high order classical spatial central, dispersion relation-preserving (DRP) (Bogey and Bailly, 2002; Brambley, 2016; Johansson, 2004) or Padé (compact) spatial discretizations (Hirsh, 1975) with summation-by-parts (SBP) operators (Strand, 1994). These high order entropy split methods not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation. The present work employs a new approach to obtain a wider class of high order entropy split methods that do not have the homogeneous property. The new method consists of a two-point numerical flux portion and a non-conservative portion in such a way that entropy conservation holds without requiring the homogeneity property of the compressible inviscid flux. For high order classical spatial central, DRP or Padé spatial discretizations, this new approach can be proven to be entropy conservative while at the same time allowing a wider class of symmetrizable inviscid flux derivatives. More importantly, we extend this new approach to derive a new entropy split method for the equations of ideal MHD. Representative test cases are illustrated with comparison among existing methods of the same high order of accuracy.

A flux-differencing formula for split-form summation by parts discretizations of non-conservative systems: Applications to subcell limiting for magneto-hydrodynamics

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. The methods are valid on unstructured curvilinear grids using tensor-product basis functions.To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks.

Rainbow Schlieren Deflectometry for spherical fields: A new algorithm and numerical validation approach

The authors present a new algorithm for analyzing ray deflection within spherical refractive index fields. The algorithm emanates from the well-known deconvolution algorithm for axisymmetric refractive index fields using the inverse Abel transformation and can be used for experimental Rainbow Schlieren Deflectometry (RSD) data. It was numerically validated by Computational Fluid Dynamics (CFD) calculations of free convection around an isothermal sphere and successive ray tracing through the determined temperature field. The algorithm determined radial temperature profiles and local Nusselt numbers from resulting deflection angles of the ray tracing simulations. An excellent agreement between original and reconstructed temperature field within the limits of assumptions was reached. This proves the usability of the algorithm for the experimental determination of spherical refractive index fields from future heat or mass transfer experiments.

A stable FDTD subgridding scheme with SBP-SAT for transient TM analysis

We proposed a provably stable FDTD subgridding method for accurate and efficient transient electromagnetic analysis. In the proposed method, several field components are properly added to the boundaries of Yee's grid to make sure that the discrete operators meet the 2nd-order and 4th-order summation-by-parts (SBP) properties. Then, by incorporating the simultaneous approximation terms (SATs) into the finite-difference time-domain (FDTD) method, the proposed FDTD subgridding method mimics the energy estimate of the continuous Maxwell's equations at the semi-discrete level to guarantee its stability. Furthermore, to couple multiple mesh blocks with different mesh sizes, the interpolation matrices are also derived. The proposed FDTD subgridding method is accurate, efficient, easy to implement and integrate into the existing FDTD codes with only simple modifications. At last, four numerical examples with fine structures are carried out to validate the effectiveness of the proposed method.

Simulation of flexural-gravity wave propagation for elastic plates in shallow water using an energy-stable finite difference method with weakly enforced boundary and interface conditions

We introduce an energy stable, high-order-accurate finite difference approximation of the dynamic, pure bending Kirchhoff plate equations for complex geometries and spatially variable properties. We utilize the summation-by-parts (SBP) framework to discretize the biharmonic operator with variable coefficients, with attention given to free and clamped boundary conditions and corner conditions. Energy conservation is established by combining SBP boundary closures with weak enforcement of the boundary and interface conditions using a penalty (simultaneous approximation term, SAT) technique. Then we couple the plate equations to the shallow water equations to study flexural-gravity wave propagation, and prove that the semi-discrete system of equations is self-adjoint. We demonstrate the stability and accuracy properties of the method on curvilinear multiblock grids using the method of manufactured solutions. The method, which we provide in an open-source code, is then used to model ocean wave interactions with the Thwaites Glacier and Pine Island Ice Shelf in the Amundsen Sea off the coast of West Antarctica.

Analysis of the fractional relativistic polytropic gas sphere

Many stellar configurations, including white dwarfs, neutron stars, black holes, supermassive stars, and star clusters, rely on relativistic effects. The Tolman–Oppenheimer–Volkoff (TOV) equation of the polytropic gas sphere is ultimately a hydrostatic equilibrium equation developed from the general relativity framework. In the modified Riemann Liouville (mRL) frame, we formulate the fractional TOV (FTOV) equations and introduce an analytical solution. Using power series expansions in solving FTOV equations yields a limited physical range to the convergent power series solution. Therefore, combining the two techniques of Euler–Abel transformation and Padé approximation has been applied to improve the convergence of the obtained series solutions. For all possible values of the relativistic parameters (σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document}), we calculated twenty fractional gas models for the polytropic indexes n = 0, 0.5, 1, 1.5, 2. Investigating the impacts of fractional and relativistic parameters on the models revealed fascinating phenomena; the two effects for n = 0.5 are that the sphere’s volume and mass decrease with increasing σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document} and the fractional parameter (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}). For n = 1, the volume decreases when σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document} = 0.1 and then increases when σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document} = 0.2 and 0.3. The volume of the sphere reduces as both σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma$$\\end{document} and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} increase for n = 1.5 and n = 2. We calculated the maximum mass and the corresponding minimum radius of the white dwarfs modeled with polytropic index n = 3 and several fractional and relativistic parameter values. We obtained a mass limit for the white dwarfs somewhat near the Chandrasekhar limit for the integer models with small relativistic parameters (α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = 1$$\\end{document}, σ=0.001\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma = 0.001$$\\end{document}). The situation is altered by lowering the fractional parameter; the mass limit increases to Mlimit = 1.63348 M⊙ at α=0.95\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = 0.95$$\\end{document} and σ=0.001\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma = 0.001$$\\end{document}.

Some Numerical Significance of the Riemann Zeta Function

In this paper, the Riemann analytic continuation formula (RACF) is derived from Euler’s quadratic equation. A nonlinear function and a polynomial function that were required in the derivation were also obtained. The connections between the roots of Euler’s quadratic equation and the Riemann Zeta function (RZF) are also presented in this paper. The method of partial summation was applied to the series that was obtained from the transformation of Euler’s quadratic equation (EQE). This led to the derivation of the RACF. A general equation for the generation of the zeros of the analytic continuation formula of the Riemann Zeta equation via a polynomial approach was also derived and thus presented in this work. An expression, which was based on a polynomial function and the products of prime numbers, was also obtained. The obtained function thus afforded us an alternative approach to defining the analytic continuation formula of the Riemann Zeta equation (ACFR). With the new representation, the Riemann Zeta function was shown to be a type of function. We were able to show that the solutions of the RACF are connected to some algebraic functions, and these algebraic functions were shown to be connected to the polynomial and the nonlinear functions. The tables and graphs of the numerical values of the polynomial and the nonlinear function were computed for a generating parameter, k, and shown to be some types of the solutions of some algebraic functions. In conclusion, the RZF was redefined as the product of a derived function, R(tn,s), and it was shown to be dependent on the obtained polynomial function.

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.