81 research outputs found
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
A Comparative Study of an Asymptotic Preserving Scheme and Unified Gas-kinetic Scheme in Continuum Flow Limit
Asymptotic preserving (AP) schemes are targeting to simulate both continuum
and rarefied flows. Many AP schemes have been developed and are capable of
capturing the Euler limit in the continuum regime. However, to get accurate
Navier-Stokes solutions is still challenging for many AP schemes. In order to
distinguish the numerical effects of different AP schemes on the simulation
results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and
the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk)
kinetic equation will be applied in the flow simulation in both transition and
continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is
used for the comparison of these two AP schemes. The numerical results show
that the UGKS captures the viscous solution accurately. The velocity profiles
are very close to the classical benchmark solutions. However, the IMEX AP
scheme seems have difficulty to get these solutions. Based on the analysis and
the numerical experiments, it is realized that the dissipation of AP schemes in
continuum limit is closely related to the numerical treatment of collision and
transport of the kinetic equation. Numerically it becomes necessary to couple
the convection and collision terms in both flux evaluation at a cell interface
and the collision source term treatment inside each control volume
Multiscale constitutive framework of 1D blood flow modeling: Asymptotic limits and numerical methods
In this paper, a multiscale constitutive framework for one-dimensional blood
flow modeling is presented and discussed. By analyzing the asymptotic limits of
the proposed model, it is shown that different types of blood propagation
phenomena in arteries and veins can be described through an appropriate choice
of scaling parameters, which are related to distinct characterizations of the
fluid-structure interaction mechanism (whether elastic or viscoelastic) that
exist between vessel walls and blood flow. In these asymptotic limits,
well-known blood flow models from the literature are recovered. Additionally,
by analyzing the perturbation of the local elastic equilibrium of the system, a
new viscoelastic blood flow model is derived. The proposed approach is highly
flexible and suitable for studying the human cardiovascular system, which is
composed of vessels with high morphological and mechanical variability. The
resulting multiscale hyperbolic model of blood flow is solved using an
asymptotic-preserving Implicit-Explicit Runge-Kutta Finite Volume method, which
ensures the consistency of the numerical scheme with the different asymptotic
limits of the mathematical model without affecting the choice of the time step
by restrictions related to the smallness of the scaling parameters. Several
numerical tests confirm the validity of the proposed methodology, including a
case study investigating the hemodynamics of a thoracic aorta in the presence
of a stent
A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for
the kinetic chemotaxis system with random inputs, which will converge to the
modified Keller-Segel model with random inputs in the diffusive regime. Based
on the generalized Polynomial Chaos (gPC) approach, we design a high order
stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time
discretization with a macroscopic penalty term. The new schemes improve the
parabolic CFL condition to a hyperbolic type when the mean free path is small,
which shows significant efficiency especially in uncertainty quantification
(UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property
will be shown asymptotically and verified numerically in several tests. Many
other numerical tests are conducted to explore the effect of the randomness in
the kinetic system, in the aim of providing more intuitions for the theoretic
study of the chemotaxis models
High order semi-implicit multistep methods for time dependent partial differential equations
We consider the construction of semi-implicit linear multistep methods which
can be applied to time dependent PDEs where the separation of scales in
additive form, typically used in implicit-explicit (IMEX) methods, is not
possible. As shown in Boscarino, Filbet and Russo (2016) for Runge-Kutta
methods, these semi-implicit techniques give a great flexibility, and allows,
in many cases, the construction of simple linearly implicit schemes with no
need of iterative solvers. In this work we develop a general setting for the
construction of high order semi-implicit linear multistep methods and analyze
their stability properties for a prototype linear advection-diffusion equation
and in the setting of strong stability preserving (SSP) methods. Our findings
are demonstrated on several examples, including nonlinear reaction-diffusion
and convection-diffusion problems
Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
We consider the development of hyperbolic transport models for the
propagation in space of an epidemic phenomenon described by a classical
compartmental dynamics. The model is based on a kinetic description at discrete
velocities of the spatial movement and interactions of a population of
susceptible, infected and recovered individuals. Thanks to this, the unphysical
feature of instantaneous diffusive effects, which is typical of parabolic
models, is removed. In particular, we formally show how such reaction-diffusion
models are recovered in an appropriate diffusive limit. The kinetic transport
model is therefore considered within a spatial network, characterizing
different places such as villages, cities, countries, etc. The transmission
conditions in the nodes are analyzed and defined. Finally, the model is solved
numerically on the network through a finite-volume IMEX method able to maintain
the consistency with the diffusive limit without restrictions due to the
scaling parameters. Several numerical tests for simple epidemic network
structures are reported and confirm the ability of the model to correctly
describe the spread of an epidemic
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