closely approach, but do not strictly reach, the cusp configuration. These
pseudocusp burst waveforms are of lower amplitude than true cusps, and follow
the same power spectrum decay of $f^{-4/3}$ at high frequencies. Accounting
for them in a search can lead to an enhancement in the number of expected
burst events.

####  6.7.4 Beyond Nambu–Goto for cusps

Gauge theory strings with energy scale $\mu_{S}^{1/2}$ have a characteristic
core width $\sim\mu_{S}^{-1/2}$ whereas classical superstrings are always
well-represented by the zero thickness Nambu–Goto limit. When a cusp forms the
string instantaneously doubles back upon itself. Ref. [1948] predicts that a
region of string around the cusp, of the order of the string thickness, will
annihilate into particles due to overlap. For strings with large tensions
(near that of the GUT scale) this modification to the cusp shape is at a small
enough scale that it does not meaningfully affect the cusp waveform. However,
as tension decreases the physical scale increases. It may prove useful to
compare the waveform of the gauge theory cusp to the superstring cusp. This
can be done by parametrically varying the core width in Abelian-Higgs
simulations.

A final assumption about the standard cusp waveform model is that it relies
only on properties universal to all strings. However, superstrings possess
additional features that could modify their cusp waveforms. Ref. [298]
suggested modifications to the amplitude based on the dimensionless loop-
length parameter $\alpha$. For example, cusps very near to Y-junctions will
have their burst amplitude modified by a correction factor that depends on the
nearby string geometry [271], while the extra degrees of freedom on
superstrings (extra dimensions) can lead to a strong suppression of both cusp
rate and burst amplitude [1949].

####  6.7.5 Cosmic strings and GW production in numerical relativity

While in most cases, perturbative methods (e.g., Nambu–Goto strings coupled to
perturbative gravity [1943]) are sufficient, situations may arise when
perturbative methods are insufficient and gravitational backreaction becomes
of ${\mathcal{}O}(1)$, even if the string tension $G\mu_{S}$ is small. For
example, a string with a sharp kink can easily lead to a locally strong field
situation [1950]. In such strong field limits, the full machinery of NR must
be employed. The construction of full NR solutions for cosmic strings was
recently achieved by [1951, 1952], where stable dynamical solutions for an
Abelian-Higgs string minimally coupled to gravity was found. This opens the
door to not just the detailed exploration of the string dynamics, but also the
characterization of the GW signals that can then be searched for.

The Abelian-Higgs model minimally coupled to gravity can be recast into the
standard BSSN/CCZ4 formalism, and solved using finite differences (we refer
the reader to [1951] for details). Presently, solutions for infinite static
strings, planar loops [1951] and single traveling wave solutions [1953] have
been constructed. Once a string configuration is constructed and successfully
evolved, the work required to extract useful GW signals from them follows very
similarly to that of any standard NR simulations, see Sec. 4.1.

As an example, in [1952], the gravitational waveform emitted during the BH
formation of a collapsing circular cosmic string was computed, and it was
shown that the waveform is dominated by the BH formation and ringdown phase,
with the primary contribution being the $\ell=2$, $m=0$ mode (as opposed to
quasi-circular compact binary merger signals which are dominated by the
$\ell=2$, $m=2$ mode). Intriguingly, due to the large asymmetric ejection of
material in the formation process, a large GW memory was also seen.

The future prospects for using NR to compute GW signals from such strong
gravity events is very promising. For example, a full general relativistic
treatment of the GW emission from cosmic string cusps and kinks is long
overdue, but is now within reach. Presently, the main obstacle to explore more
complicated string configurations is the need for the construction of accurate
initial conditions – a usual problem in any NR endeavor, here made more
difficult by the presence of fundamental matter fields. Nevertheless, this
area is ripe for further exploration.

##  7 Afterword

LISA will usher in an era of millihertz GW astronomy that will open a window
to new classes of sources and offer unprecedented opportunities to probe our
understanding of the universe. Integral to this scientific vision are
predictions of the source waveforms, which are necessary for the detection and
interpretation of GW events.

This white paper has analyzed the question of waveform preparedness in the
LISA era. It has reviewed the modelling requirements from astrophysical and
data analysis perspectives, and has analyzed the status of the main approaches
to modelling waveforms from compact binaries both in vacuum GR and in the
presence of environmental influences or effects from new physics beyond GR and
the Standard Model. This white paper also provides guidance on new
developments that are needed for the different approaches, both in terms of
modelling accuracy and the covered parameter space, so that gravitational wave
models will be ready for the LISA era.

The experience with ground-based GW detectors has to some extent prepared the
community. For example, several approaches that predict the waveforms for
comparable mass black hole binaries can be applied directly to massive black
hole binaries due to the simple mass scaling of GR. However, these models are
not currently sufficient to fully realize LISA’s science goals. In particular
they insufficiently cover the parameters of eccentricity, highly asymmetric
mass-ratios, and high spin magnitudes. In addition, the possibility of very
strong massive black hole binary signals places additional demands on the
accuracy of these models, well beyond what can currently be achieved.

For extreme-mass-ratio inspirals, leading-order (adiabatic) models that are
sufficient to enable some LISA science are becoming available, and these
models can be evaluted fast enough for use in LISA data analysis. In order to
enable the full spectrum of LISA science with EMRIs post-adiabatic models are
needed. These have recently been developed for the simplest orbital
configurations (non-spinning, quasi-circular). It also looks likely that these
models can cover much of the intermediate mass ratio parameter space.
Significant work remains to extend these post-adiabatic models to cover the
full parameter space of precessing and eccentric parameter binaries where both
components are spinning.

The modelling of sources in theories beyond GR is still in its infancy with
significant effort needed. For example, the majority of computations have
focused on higher derivative gravity, that are most interesting in the high
curvature regime, or on Horndeski models. Moreover, simulations of compact
binaries in extensions of GR are, at the time of writing this white
paper,proof-of-concept. That is, they have been done for nearly equal-mass
binaries, typically of non spinning black holes, and at relatively low
resolution. This is comparable to the state of numerical relativity in GR
about $15$ years ago (i.e., few years after the breakthrough in 2005).
Likewise, perturbative or PN calculations have only started to go beyond the
leading order in the correction. To perform theory-specific tests of gravity,
more work is needed, both in terms of modelling accuracy and in covering
(parts of) the parameter space of black hole masses, spins and additional
theory-specific parameters.

The good news is that in all approaches there are clear ideas of what needs to
be done in order to reach the modelling requirements for LISA. Doing so will
require a concerted effort over the next decade. Given sufficient resources,
we are positive that all goals can be achieved.

We particularly recommend that the community conduct more investigations into
the waveform standards and waveform interface. While lessons have been learned
from ground-based detectors, LISA will offer unique challenges that will need
to be addressed by the waveform and data analysis communities working
together.

##  Appendix A Descriptions of NR codes

Description of some numerical relativity codes that focus, in particular, on
modelling LISA sources. A more extensive list is given in Table 7.

BAM

BAM is a modular code framework initially developed at the University of Jena
and is now maintained and further developed by the CoRe (Computational
Relativity) Collaboration [616, 582, 617, 618]. BAM employs finite differences
for the discretization of spacetime fields, and high-resolution shock-
capturing methods for the evolution of general relativistic hydrodynamics
variables. It features automated code generation with Mathematica, adaptive
mesh refinement (AMR) in space and time, and parallelization with MPI and
OpenMP for large scale HPC, recently at 70% on up to 50k cores. BAM is
employed for the simulation of compact binary mergers, e.g., [524, 1910, 538,
1954]. A major focus is on binary black hole, binary neutron star, and black
hole - neutron star systems [1955], plus more exotic compact objects or merger
scenarios [1685]. Recent updates allow the study of radiation hydrodynamics
[1956], magnetohydrodynamics, and the usage of an entropy-viscosity limiter
for the flux computation [1957]. BAM has contributed numerical data for the
development and validation of e.g. Phenom-type and EOB waveform models.
Several hundred binary neutron star merger simulations are available as part
of the existing CoRe database [619, 620].

BAMPS

The Jena-Lisbon collaboration is developing the code bamps, a new and highly
scalable NR code for exascale applications using pseudo-spectral and DG
methods [622, 621]. A key feature is hp-refinement for spectral element
methods [623], which allows highly scalable and accurate simulations,
currently with a focus on smooth fields. To date, bamps has already been
employed for the study of GW collapse [622, 1958], simulations with scalar
fields for collapse and for boson stars are in development. Further
applications include simple neutron star spacetimes [621], and studies of new
techniques regarding the dual foliation framework [1959].

Dendro-GR

Dendro-GR [626, 1960, 1961] uses an unstructured grid with a localized,
wavelet-based refinement scheme to evolve the BSSN formulation of the Einstein
equations using moving punctures. In test simulations of a BBH with a mass
ratio of $10:1$, Dendro-GR shows good scaling to over $10^{5}$ cores.

Einstein Toolkit

The Einstein Toolkit [628, 629] is an open-source cyberinfrastructure for
computational astrophysics, with more than $300$ registered users worldwide.
It implements Einstein’s equations, or extensions thereof, in the BSSN or CCZ4
formulation together with the moving puncture gauge. It employs the method of
lines, in which spatial derivatives are implemented as up to $8^{\mathrm{t}h}$
order finite differences together with a collection of direct time integration
techniques (e.g., fourth order Runge-Kutta integrator). The Einstein Toolkit
is based on the Cactus computational toolkit [1962] and the Carpet boxes-in-
boxes AMR package [1963, 1964]. It also provides a multipatch infrastructure
consisting of Cartesian boxes-in-boxes meshes in the region where the binaries
evolve (typical size $\sim 50\dots 100M$), and a spherical outer region in the
wave zone (typical radius $\sim 1000\dots 2000M$) provided by the Llama code.
Kreiss-Oliger dissipation is employed to reduce high frequency noise at
refinement boundaries. The toolkit uses hybrid MPI/OpenMP parallelization. The
Einstein Toolkit has been a broad community project with a multitude of
software that is based on its infrastructure or that has been developed within
its framework. The additional software is indicated by asterisks in Table 7.

Einstein Toolkit – CarpetX

The Einstein Toolkit consortium is developing a new AMR driver, CarpetX
[1965], for the Einstein Toolkit. CarpetX leverages the AMReX framework [1966]
to provide scalable adaptive mesh refinement for physics modules. CarpetX will
provide: efficient support for both CPUs and GPUs; parallelization via MPI and
OpenMP; SIMD vectorization; scalable I/O based on the ADIOS2 file format
[1967] and the openPMD metadata standard; and more. CarpetX offers block-
structured AMR based on local error estimates, exact conservation across mesh
interfaces, higher order prolongation operators, and scalable elliptic solvers
based on PETSc [1968]. CarpetX is available as open source and was used in
production in Ref. [1969].

GR-Athena++

Another example is GR-Athena++[652] based on the astrophysical (radiation)
magnetohydrodynamics code Athena++ [1970]. GR-Athena++ leverages on Athena++’s
oct-tree AMR and exploits hybrid parallelism at different levels. The standard
distributed approach is augmented by a dynamical tasklist for the procedures
in each mesh block that overlaps calculation and communication (thus
mitigating some of the overhead associated with high order finite
differencing). In-core vectorization and other optimization techniques of the
basic kernels [1971] are also employed. This approach leads to a high parallel
efficiency, with strong scaling efficiencies above 95% for up to
$10^{4}\,$CPUs and excellent weak scaling is up to $10^{5}\,$CPUs measured in
a production BBH setup with AMR.

GRChombo

GRChombo ([653, 655, 654]) is an open-source NR code built using an optimized
version of the publicly available library Chombo[1972] developed at LBNL. It
is based on established methods of solving the Einstein equations, but the
highly flexible AMR capability and templating over physics classes makes it
suited to strong gravity problems in beyond-GR scenarios (e.g. Horndeski
gravity [1973], cosmic strings [1952]), which have been key targets of the
code. GRChombo is written entirely in C++14, using hybrid MPI/OpenMP
parallelism and vector intrinsics (in particular, the evolution calculations
are explicitly vectorised) to achieve improved performance on the latest
architectures. It has good strong scaling up to around 4,000 cores for typical
binary BH problems, at which point it becomes limited by problem size as with
most traditional codes.

The next release of Chombo is designed to be performance portable to different
heterogeneous architectures (including GPUs), which GRChombo should be able to
leverage to improve scaling further.

Illinois GRMHD

The Illinois GRMHD evolution code [1974, 659, 660] solves the Einstein field
equations via the BSSN scheme with puncture gauge conditions, utilizes a
conservative high-resolution, shock-capturing (HRSC) scheme for the matter and
adapts the Carpet infrastructure to implement AMR. The code solves the
magnetic induction equation by introducing a magnetic vector potential, which
guarantees that the B-field remains divergence-free for any interpolation
scheme used on refinement level boundaries. This formulation reduces to a
standard constrained transport (CT) scheme on uniform grids. An improved
Lorenz gauge condition for evolving the vector potential in AMR without the
appearance of spurious B-fields on refinement level boundaries is implemented.
This gauge exhibits no zero-speed modes, enabling spurious magnetic effects to
propagate off the grid quickly. The GRMHD code recently added a radiation
module built to handle transport (photons or neutrinos) via the M1 moment
formalism. A version of the Illinois GRMHD code has been released as an open-
source module that has been ported to the Einstein Toolkit, where it has been
documented and rewritten to make it more user-friendly, modular and efficient.
The code has been used to simulate numerous scenarios related to LISA sources
(e.g., BHBH mergers, BHBH mergers in gaseous clouds and magnetized disks,
etc.) and their associated gravitational waveforms.

MHDuet

The publicly available code MHDuet [663, 664, 1975, 1976] is generated by the
open-source platform Simflowny to run under the SAMRAI infrastructure, which
provides the parallelization and the adaptive mesh refinement. There are
different versions of the code, either to study alternative gravity theories
or to include large-eddy-simulations and neutrino transport.

SACRA

The SimulAtor for Compact objects in Relativistic Astrophysics (SACRA) code
solves Einstein’s equation with the BSSN-puncture formulation together with a
Z4c constraint propagation prescription and the relativistic hydrodynamics
equation with the HRSC scheme [666, 667]. It implements the box-in-box
conservative AMR with parallelization by MPI and OpenMP. The code has been
used to simulate compact binary mergers such as NSNS mergers, BHNS mergers,
and BHBH mergers. The NSNS waveform catalog SACRA Gravitational Wave Data Bank
is available at [671]. The latest version of SACRA has a couple of branches,
such as SACRA-TD, to simulate the tidal disruption of an ordinary star by the
SMBH [1977], or SACRA-MG to simulate a compact object in scalar-Gauss-Bonnet
theory [668] and scalar-tensor theory [669]. SACRA also has a version to solve
high-dimensional Einstein’s equation [670]. In addition, it features modules
for neutrino radiation transport [1978], MHD [1979, 1980] and viscous
hydrodynamics [1981].

SACRA-SFS2D

is a viscous-radiation hydrodynamics [672] and radiation magnetohydrodynamics
code [673] in full general relativity. These codes can be applied to the
collapse of massive stars and supermassive stars to a stellar-mass and
supermassive black hole [1982], to the post-merger evolution of the neutron-
star binaries [1983], and to collapsar modeling [1984].

SpEC

The Spectral Einstein Code (SpEC) [674] is a pseudo-spectral GR code developed
by the Simulating eXtreme Spacetimes collaboration. SpEC is capable of
evolving BBH and binaries with neutron stars. It incorporates an elliptic
solver to construct initial data [530, 534, 537], eccentricity reduction
[1985, 1986, 1987], constraint-preserving and non-reflective outer boundary
conditions [544, 1988, 1989], and implements hp-AMR [744, 1990]. Extracted
gravitational waves are extrapolated to future null infinity [556, 1991, 523]
and corrected for center-of-mass drift [1992]. SpEC is a highly accurate NR
code for BBH, and has computed large waveform catalogs of BBH inspirals
encompassing several tens of orbits [521, 523]. SpEC has evolved BBH with
spins as large as 0.994 [1993], inspirals as long as 170 orbits [584] as well
as eccentric binaries [717].

SpECTRE

The Simulating eXtreme Spacetimes collaboration is developing the next-
generation code SpECTRE [677, 676], which uses a discontinuous-Galerkin-
finite-difference hybrid method [1994, 1995, 1996] for accurate and robust
neutron star simulations combined with task-based parallelism, showing good
scaling to over $600,000$ cores [676]. Using Cauchy-Characteristic extraction,
SpECTRE is able to resolve gravitational wave memory [560, 561] and improve
hybridization with post-Newtonian waveforms by ensuring numerical relativity
waveforms are in the same BMS frame as the post-Newtonian waveforms [563].
Improvements for efficiently simulating binary black holes with mass ratios
$q>10$ aim to reduce the computational cost of such simulations by a factor of
$q$ [695]. SpECTRE also incorporates a flexible elliptic solver [1997, 1998,
541].

Spritz

Spritz is an open-source code that solves the equations of general
relativistic magnetohydrodynamics and that can take into account finite
temperature nuclear equations of state and neutrino emission [643, 1999,
2000]. The code is based on the Einstein Toolkit framework and implements also
high-order methods for the hydrodynamic equations.

WhiskyMHD

WhiskyMHD is a fully general relativistic magnetohydrodynamic code based on
the Einstein Toolkit [648]. The code has been used mainly to perform
simulations of binary neutron star mergers, but it has also been used to
perform the first simulations of magnetized plasma around merging supermassive
black holes in the ideal magnetohydrodynamic approximation [2001].

## Acknowledgements

We would like to thank our internal reviewers, Leor Barack, Christopher Berry,
and Nelson Christensen for extremely helpful feedback. Sarp Akçay acknowledges
University College Dublin’s Ad Astra Fund. Pau Amaro Seoane acknowledges the
funds from the “European Union NextGenerationEU/PRTR”, Programa de Planes
Complementarios I+D+I (ref. ASFAE/2022/014). Enrico Barausse acknowledges
support from the European Union’s H2020 ERC Consolidator Grant “GRavity from
Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673) and the EU
Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-
Curie Grant Agreement No. 101007855. Emanuele Berti is supported by NSF Grants
No. AST-2006538, PHY-2207502, PHY-090003 and PHY-20043; by NASA Grants No.
20-LPS20-0011 and 21-ATP21-0010; by the John Templeton Foundation Grant 62840;
and by the ITA-USA Science and Technology Cooperation program, supported by
the Ministry of Foreign Affairs of Italy (MAECI). Richard Brito acknowledges
financial support provided by FCT/Portugal under the Scientific Employment
Stimulus – Individual Call – 2020.00470.CEECIND and under project No.
2022.01324.PTDC. Marta Colleoni, Sascha Husa, Anna Heffernan and Pierre
Mourier are supported by the Spanish Agencia Estatal de Investigaciòn grants
PID2022-138626NB-I00, PID2019-106416GB-I00, IJC2019-041385, funded by
MCIN/AEI/10.13039/501100011033; the MCIN with funding from the European Union
NextGenerationEU/PRTR (PRTR-C17.I1); Comunitat Autonòma de les Illes Balears
through the Direcció General de Recerca, Innovació I Transformació Digital
with funds from the Tourist Stay Tax Law (PDR2020/11 - ITS2017-006), the
Conselleria d’Economia, Hisenda i Innovació grant numbers SINCO2022/18146 and
SINCO2022/6719, co-financed by the European Union and FEDER Operational
Program 2021-2027 of the Balearic Islands; the “ERDF A way of making Europe”;
Alvin J. K. Chua and Jonathan E. Thompson acknowledge support from the NASA
LISA Preparatory Science grant 20-LPS20-0005. Katy Clough acknowledges funding
from the UKRI Ernest Rutherford Fellowship (grant number ST/V003240/1). David
A. Nichols acknowledges support from the NSF Grants No. PHY-2011784 and No.
PHY-2309021. Vasileios Paschalidis is supported by NSF Grant No. PHY-2145421
and NASA Grant No. 80NSSC22K1605. Stuart Shapiro is supported by NSF Grants
No. PHY-2006066 and PHY-2308242. Scott Field acknowledges support from US
National Science Foundation Grants Nos. PHY-2110496, DMS-2309609, and by UMass
Dartmouth’s Marine and Undersea Technology (MUST) Research Program funded by
the Office of Naval Research (ONR) under Grant No. N00014-23-1–2141. Davide
Gerosa is supported by ERC Starting Grant No. 945155–GWmining, Cariplo
Foundation Grant No. 2021-0555, MUR PRIN Grant No. 2022-Z9X4XS, Leverhulme
Trust Grant No. RPG-2019-350, MSCA Fellowship No. 101064542–StochRewind, and
the ICSC National Research Centre funded by NextGenerationEU. Eliu Huerta
acknowledges support from the U.S. Department of Energy under Contract No. DE-
AC02-06CH11357, and from the U.S. National Science Foundation (NSF) through
award OAC-2209892. Scott A. Hughes has been supported by NASA ATP Grant
80NSSC18K1091, NSF Grants PHY-1707549 and PHY-2110384, by MIT’s Margaret
MacVicar Faculty Fellowship Program, and by the MIT Kavli Institute for
Astrophysics and Space Research. Chris Kavanagh acknowledges support from
Science Foundation Ireland under Grant number 21/PATH-S/9610. Gaurav Khanna
acknowledges support from US National Science Foundation Grants No.
PHY-2307236 and DMS-2309609. Larry Kidder acknowledges support from NSF
OAC-2209655, PHY-2308615 and Sherman Fairchild Foundation. Pablo Laguna
acknowledges support from US National Science Foundation Grants No.
PHY-2114582 and 2207780. Georgios Lukes-Gerakopoulos has been supported by the
fellowship Lumina Quaeruntur No. LQ100032102 of the Czech Academy of Sciences.
Hyun Lim is supported by the LANL Laboratory Directed Research and Development
program under project number 20220087DR. LANL is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of the U.S.DOE
(Contract No. 89233218CNA000001). This work is authorized for unlimited
release under LA-UR-23-31548. Tyson B. Littenberg is supported by the NASA
LISA Study Office. Carlos O. Lousto gratefully acknowledge the National
Science Foundation (NSF) for financial support from Grant No. PHY- 1912632 and
PHY-2207920. Elisa Maggio acknowledges funding from the Deutsche
Forschungsgemeinschaft (DFG) - project number: 386119226. Richard
O’Shaughnessy acknowledges support from NSF PHY-2012057, PHY-2309172, and
AST-2206321. Naritaka Oshita is supported by the Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant Number JP23K13111. Rodrigo P. Macedo
and Maarten van de Meent acknowledge the financial support by the VILLUM
Foundation (grant No. VIL37766), the DNRF Chair program (grant No. DNRF162) by
the Danish National Research Foundation, and the European Union’s H2020 ERC
Advanced Grant “Black holes: gravitational engines of discovery” grant
agreement No. Gravitas–101052587. Adam Pound acknowledges the support of a
Royal Society University Research Fellowship and a UKRI Frontier Research
Grant under the Horizon Europe Guarantee scheme [grant number EP/Y008251/1].
Milton Ruiz acknowledges support from the Generalitat Valenciana Grant
CIDEGENT/2021/046 and the Spanish Agencia Estatal de Investigaci´on Grant
PID2021-125485NB-C21. Carlos F. Sopuerta CFS is supported by contracts
PID2019-106515GB-I00 and PID2022-137674NB-I00 (MCIN/AEI/10.13039/501100011033)
and partially supported by the program Unidad de Excelencia María de Maeztu
CEX2020-001058-M (MCIN/AEI/10.13039/501100011033). Stuart L. Shapiro
acknowledges support from NSF Grants No. PHY-2006066 and No. PHY-2308242.
Deirdre Shoemaker acknowledges support from NASA 22-LPS22-0023, 80NSSC21K0900
and NSF 2207780. Antonios Tsokaros is supported by NSF Grants No. PHY-2308242
and OAC-2310548. Niels Warburton acknowledges support from a Royal Society -
Science Foundation Ireland University Research Fellowship. This publication
has emanated from research conducted with the financial support of Science
Foundation Ireland under Grant numbers 16/RS-URF/3428, 17/RS-URF-RG/3490 and
22/RS-URF-R/3825. Helvi Witek acknowledges support provided by the National
Science Foundation under NSF Awards No. OAC-2004879 and No. PHY-2110416. Huan
Yang is supported by the Natural Sciences and Engineering Research Council of
Canada and in part by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development Canada and by
the Province of Ontario through the Ministry of Colleges and Universities.
Miguel Zilhão acknowledges financial support by the Center for Research and
Development in Mathematics and Applications (CIDMA) through the Portuguese
Foundation for Science and Technology (FCT – Fundação para a Ciência e a
Tecnologia) – references UIDB/04106/2020 and UIDP/04106/2020 – as well as FCT
projects 2022.00721.CEECIND and 2022.04560.PTDC.

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