arXiv subjects

Sebastiano Boscarino

Publications and source records attributed to Sebastiano Boscarino.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic-Preserving Exponential Integrators applied to the Hyperbolic Korteweg-de Vries System

We study the application of exponential time integration methods, coupled with Fourier pseudospectral space discretization, to the hyperbolic Korteweg-de Vries (KdVH) system. We investigate the asymptotic preserving (AP) properties of such discretizations, showing that, in general, Lawson methods are not asymptotic preserving, because the auxiliary (derivative-approximating) variables do not satisfy the limit equilibrium manifold, whereas exponential time differencing (ETD) methods are AP for all components, including both the solution variable and the auxiliary variables. We also present an efficient numerical implementation based on an exact formula for the matrix exponential of this system, and compare it with previously-proposed ImEx Runge-Kutta time integration, showing that exponential methods can be competitive in this context.

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arXiv2026

Asymptotic preserving scheme for the shallow water equations with non-flat bottom topography and Manning friction term

In our previous work [29], we proposed a class of high-order asymptotic preserving (AP) finite difference weighted essentially non-oscillatory (WENO) schemes for solving the shallow water equations (SWEs) with bottom topography and Manning friction, utilizing a penalization technique inspired by [6]. Although the added weighted diffusive term enhanced stability, it increased computational cost and slowed down the convergence rate in the intermediate regime between convection and diffusion. In this paper, we extend our previous study by removing the penalization while preserving the AP property. To achieve this, we employ a high order semi-implicit implicit-explicit Runge-Kutta (SI-IMEX-RK) time discretization, coupled with the high-order WENO reconstruction for first-order derivatives and a central difference scheme for second-order spatial derivatives. This combination yields a class of fully high-order schemes. Theoretical analysis and numerical experiments demonstrate that the proposed schemes retain AP, asymptotically accurate (AA) and well-balanced properties, while offering higher computational efficiency compared to our previous schemes in [29], especially in the intermediate regime between convection and diffusion. Moreover, treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise, the scheme fails to converge to the limiting equations. This indicates that implicit treatment of Manning friction is necessary for the stability of the method.

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arXiv2026

Asymptotic Analysis of IMEX-RK Methods for ES-BGK Model at Navier-Stokes level

Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further suggest some IMEX-RK methods developed in the literature to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.

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arXiv2025

A quantitative comparison of high-order asymptotic-preserving and asymptotically-accurate IMEX methods for the Euler equations with non-ideal gases

We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK approach involves an implicit coupling between the momentum and energy balance so as to avoid the acoustic CFL restriction, while the density can be treated in a fully explicit fashion. This approach leads to a mildly nonlinear equation for the pressure, which can be solved according to a fixed point procedure. An alternative strategy consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases. The spatial discretization is based on a Discontinuous Galerkin (DG) method, which naturally allows high-order accuracy. The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases.

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arXiv2025

A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

In this paper, we propose a high order conservative semi-Lagrangian scheme (SL) for the ellipsoidal BGK model of the Boltzmann transport equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator with small Knudsen number, we employ high order $L$-stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit schemes are designed to update solutions explicitly without resorting to any Newton solver. We present several numerical tests to demonstrate the accuracy and efficiency of the proposed methods. These methods allow us to obtain accurate approximations of the solutions to the Navier-Stokes equations or the Boltzmann equation for moderate or relatively large Knudsen numbers, respectively.

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arXiv2025

Approximate Taylor methods for ODEs

A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge-Kutta schemes. Compared to Runge-Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansion based on exact derivatives. The numerical results that are obtained with our proposal are satisfactory and show that this approximate approach can attain results as good as the exact Taylor procedure with less implementation and computational effort.

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arXiv2023

High order asymptotic preserving and well-balanced schemes for the shallow water equations with source terms

In this study, we investigate the Shallow Water Equations incorporating source terms accounting for Manning friction and a non-flat bottom topology. Our primary focus is on developing and validating numerical schemes that serve a dual purpose: firstly, preserving all steady states within the model, and secondly, maintaining the late-time asymptotic behavior of solutions, which is governed by a diffusion equation and coincides with a long time and stiff friction limit. Our proposed approach draws inspiration from a penalization technique adopted in {\it{[Boscarino et. al, SIAM Journal on Scientific Computing, 2014]}}. By employing an additive implicit-explicit Runge-Kutta method, the scheme can ensure a correct asymptotic behavior for the limiting diffusion equation, without suffering from a parabolic-type time step restriction which often afflicts multiscale problems in the diffusive limit. Numerical experiments are performed to illustrate high order accuracy, asymptotic preserving, and asymptotically accurate properties of the designed schemes.

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arXiv2022

Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws

The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in a previous paper for explicit schemes based on the application of a well-balanced reconstruction operator has been applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws.

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arXiv2021

High Order Semi-implicit WENO Schemes for All Mach Full Euler System of Gas Dynamics

In this paper, we propose a new high order semi-implicit scheme for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. Type A IMEX schemes are constructed to handle not well-prepared initial conditions. Besides, these schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes for well-prepared initial conditions. Divergence-free property of the time-discrete schemes is proved. The proposed scheme can also well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes.

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arXiv2021

A local velocity grid conservative semi-Lagrangian schemes for BGK model

Most numerical schemes proposed for solving BGK models for rarefied gas dynamics are based on the discrete velocity approximation. Since such approach uses fixed velocity grids, one must secure a sufficiently large domain with fine velocity grids to resolve the structure of distribution functions. When one treats high Mach number problems, the computational cost becomes prohibitively expensive. In this paper, we propose a velocity adaptation technique in the semi-Lagrangian framework for BGK model. The velocity grid will be set locally in time and space, according to mean velocity and temperature. We apply a weighted minimization approach to impose conservation. We presented several numerical tests that illustrate the effectiveness of our proposed scheme.

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arXiv2021

BGK models for inert mixtures: comparison and applications

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

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arXiv2020

Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures

In this work, we propose a class of high order semi-Lagrangian scheme for a general consistent BGK model for inert gas mixtures. The proposed scheme not only fulfills indifferentiability principle, but also asymptotic preserving property, which allows us to capture the behaviors of hydrodynamic limit models. We consider two hydrodynamic closure which can be derived from the BGK model at leading order: classical Euler equations for number densities, global velocity and temperature, and a multi-velocities and temperatures Euler system. Numerical simulations are performed to demonstrate indifferentiability principle and asymptotic preserving property of the proposed conservative semi-Lagrangian scheme to the Euler limits.

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arXiv2020

High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time-stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

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arXiv2020

Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction

In this paper, we propose and analyse a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper.

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arXiv2020

Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, 2020]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to rigid body rotation, Vlasov Poisson system and the BGK model of rarefied gas dynamics. In the first two cases operator splitting is adopted, to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model.

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arXiv2020

Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules

In this paper, we propose a new semi-Lagrangian scheme for the polyatomic ellipsoidal BGK model. In order to avoid time step restrictions coming from convection term and small Knudsen number, we combine a semi-Lagrangian approach for the convection term with an implicit treatment for the relaxation term. We show how to explicitly solve the implicit step, thus obtaining an efficient and stable scheme for any Knudsen number. We also derive an explicit error estimate on the convergence of the proposed scheme for every fixed value of the Knudsen number.

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arXiv2019

High order conservative Semi-Lagrangian scheme for the BGK model of the Boltzmann equation

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

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arXiv2018

On the stability of Approximate Taylor methods for ODE and their relationship with Runge-Kutta schemes

In [Baeza et al., Computers and Fluids, 159, 156--166 (2017)] a new method for the numerical solution of ODEs is presented. This methods can be regarded as an approximate formulation of the Taylor methods and it follows an approach that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their high order derivatives, are needed. In this reference, the absolute stability region of the new methods is conjectured to be coincident with that of their exact counterparts. There is also a conjecture about their relationship with Runge-Kutta methods. In this work we answer positively both conjectures.

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