Research activity. Numerical methods for hyperbolic problems

Hyperbolic systems of balance laws pose several challenging mathematical and numerical problems. Numerical difficulties arise mainly in presence of nonlinear fluxes which may originate shocks and stiff source terms which may pose severe time step restrictions. The crucial point is to derive schemes able to capture the relevant structure of the solutions such as singularities, shocks, instabilities without resolving the small space and time scales.

The research activity concerns different (related) subjects:
  1. Asymptotic preserving schemes for diffusion limits
  2. Implicit-Explicit schemes for time dependent problems
  3. Central schemes for hyperbolic balance laws
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1. Asymptotic Preserving schemes for diffusion limits
Even if it is difficult to give a rigorous definition of Asymptotic Preserving (AP) scheme since the concept has been used for a long time in the physics and mathematics literature and may refer to different discretization parameters we can formalize the notion as follows. Consider a singular perturbation problem Peps whose solutions converge to those of a limit problem P0 when the perturbation parameter 'eps' tends to zero. A scheme Peps,h for problem Peps with discretization parameters 'h' is called Asymptotic Preserving if its stability requirement on h is independent of 'eps' and if its limit P0,h when 'eps' tends to zero is consistent with the limit problem P0.
 Figure 1: The AP-diagram

Prototype examples of such a situation are given by differential algebraic equations (DAE) in ODEs and hyperbolic relaxation systems in PDEs. The study of the zero relaxation limit for such systems has caught much interest, both from a theoretical and numerical point of view (Chen, D.Levermore, T-P.Liu, Comm. Pure App. Math., 1992, [pdf file], S.Jin, Z.Xin, Comm. Pure App. Math. 1995, [pdf file]).
In this research topic we are concerned with numerical methods for hyperbolic systems with relaxation under diffusive scaling, usually called hyperbolic systems with diffusive relaxation. In such a case the usual IMEX discretizations for hyperbolic relaxation systems (see next paragraph on IMEX schemes) do not work , because in the diffusive scaling characteristic speeds diverge when the relaxation parameter vanishes and the asymptotic behavior of the systems is governed by a reduced parabolic system. In rarefied gas dynamics this scaling leads from the Boltzmann equation to the incompressible Navier-Stokes equations.
 Figura 2. Solution of the semiconductor Boltzmann equation by diffusive relaxation schemes

Using a suitable reformulation of the problem we have been able to derive numerical schemes, referred as diffusive relaxation schemes, that can work uniformly with respect to the diffusive relaxation parameter and that can be applied to a large class of systems [1-3] including kinetic equations [4, 5]. A closely related approach can be used to treat the case where the asymptotic behavior is characterized by fourth-order diffusion equations [6] (follow this link for some animations).

References
  1. S.Jin, L.Pareschi, G.Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numerical Analysis, Vol. 35, No. 6, pp. 2405-2439, (1998). 
  2. G.Naldi, L.Pareschi, Numerical schemes for kinetic equations in diffusive regimes, Applied Math. Letters, Vol.11, No.2, pp. 29-35, (1998). 
  3. G.Naldi, L.Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM J. Numerical Analysis, Vol. 37, No. 4, pp. 1246-1270, (2000).
  4. S.Jin, L. Pareschi, G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numerical Analysis, Vol. 38, No. 13, pp. 913-936, (2000).
  5. S.Jin, L.Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comp. Phys. 161, pp.312-330, (2000).
  6. G.Naldi, L.Pareschi, G.Toscani, Relaxation schemes for PDEs and applications to fourth order diffusion equations, Surveys on Mathematics for Industry, 10, pp.315-343, (2002).
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    2. Implicit-Explicit schemes for time dependent problems
    Several physical phenomena of great importance for applications are described by large stiff systems of differential equations of the form
    where F is a non stiff term and G a stiff term.
    Such systems include different time scales and may represent a system of ODE's or a discretization of a system of PDE's, such as, for example, convection-diffusion equations, reaction-diffusion equations and hyperbolic systems with relaxation. In this research we focused on the latter case, which in recent years has been a very active field of research, due to its great impact on applied sciences. We mention that hyperbolic systems with relaxation appears in kinetic theory of rarefied gases, hydrodynamical models for semiconductors, viscoelasticity, multiphase flows and phase transitions, radiation hydro-dynamics, traffic flows, shallow waters, etc.
    Figure 1. Boundaries of stability regions for some IMEX schemes.

    The development of efficient numerical schemes for such systems is challenging, since in many applications the relaxation time which characterizes the stiffness of the system varies from values of order one to very small values if compared to the time scale determined by the characteristic speeds of the system.
    In this second case the hyperbolic system with relaxation is said to be stiff and typically its solutions are well approximated by solutions of a suitable reduced set of conservation laws called equilibrium system. Usually it is extremely difficult, if not impossible, to split the problem in separate regimes and to use different solvers in the stiff and non stiff regions. Thus one has to use the original relaxation system in the whole computational domain. In our research we have constructed high order, underresolved Runge-Kutta time discretization for such systems. In particular, using the formalism of Implicit-Explicit (IMEX) Runge-Kutta schemes (U.Asher, S.Ruth, B.Wetton, SINUM, 1995, [pdf file]) we derived new IMEX schemes up to order 3 that are strong-stability-preserving (SSP) for the limiting system of conservation laws [1, 2, 4].

    Figure 2: Density contours for the double Mach reflection problem

    To this aim, we derived general conditions that guarantee the asymptotic preserving (AP) property, i.e. the consistency of the scheme with the equilibrium system and the asymptotic accuracy, i.e. the order of accuracy is maintained in the stiff limit. Applications of IMEX schemes to compressible and incompressible fluid-limits have been considered in [3, 5]. Recently we have extended IMEX schemes to treat the case of diffusion limits [6] (see above paragraph).

    References
    1. L.Pareschi, G.Russo Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations, Recent Trends in Numerical Analysis, Edited by L.Brugnano and D.Trigiante, Vol. 3, 269-289, (2000).
    2. L.Pareschi, G.Russo, Asimptotically SSP schemes for hyperbolic systems with stiff relaxation, Hyperbolic Problems: Theory, Numerics, Applications, Proceedings of the Ninth International Conference on Hyperbolic Problems Held in Caltech, Pasadena, March 25-29, 2002, Springer, (2003), pp.241-255.
    3. M.K.Banda, A.Klar, L.Pareschi, M.Seaid, Compressible and Incompressible Limits for Hyperbolic Systems with Relaxation, Journal of Computational and Applied Mathematics, 168 (2004) pp.41-52.
    4. L.Pareschi, G.Russo, Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25 (2005), no. 1-2, 129-155.
    5. Banda, Mapundi ; Klar, Axel ; Pareschi, Lorenzo ; Seaïd, Mohammed . Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations. Math. Comp. 77 (2008), no. 262, 943-965.
    6. Boscarino S., Pareschi L, Russo G., IMEX Runge-Kutta schemes and hyperbolic systems of conservation laws with stiff diffusive relaxation, ICNAAM, AIP Conference Proceedings 1168, (2009) pp.1106-1111.
    7. Boscarino S., Pareschi L, Russo G., Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, preprint

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    3. Central schemes for hyperbolic balance laws
    In the last few decades systems of balance laws 
    have been an active field of research, both for the challenging mathematical and numerical problems they pose, and for the numerous applications in various sectors such as gas dynamics, mathematical modeling of semiconductor devices, or shallow water models for wave propagation in lakes and rivers. Besides the case of stiff sources and/or fluxes already discussed additional difficulties are related to the computation of steep gradients (shocks) and small signals which can be lost by the numerical dissipation of the scheme. 
    Modern high order shock-capturing finite volume methods for hyperbolic systems of conservation laws are based on a conservative discretization of the equations, in which the unknown represents the cell average of the solution; such discretization guarantees the correct speed of propagation of the discontinuities.
     Figure 1: The staggered grid of central schemes

    Central schemes (H. Nessyahu, E. Tadmor, J. Comp. Phys., 1990, [pdf file])  are based on a staggered discretization in space and offer a ``black-box'' solver for a wide variety of problems governed by multi-dimensional systems of non-linear conservation laws and related equations. Central Runge-Kutta (CRK) schemes [4] represent an high-order extension of standard secondo-order central schemes and are based on a staggered discretization in space, in which the numerical solution of the Runge-Kutta scheme is obtained by a conservative scheme, while the so called stage-values are computed by a non conservative predictor.
    Figure 2: Solution of Lax problem by CRK schemes

    In this context we also considered extension of the central schemes to include stiff sources using a predictor-corrector strategy to achieve uniform accuracy [1, 2, 3].

    References
    1. E.Gabetta, L.Pareschi, M.Ronconi, Central schemes for hydrodynamical limits of discrete-velocity kinetic equations, Transp. Theo. Stat. Phys. 29, 3-5, pp.465-477, (2000).
    2. L.Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms, SIAM J. Numer. Anal. 39 (2001), no. 4, 1395--1417.
    3. S.Jin, L.Pareschi, M.Slemrod, A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansion, Acta Mathematicae Applicatae Sinica, Vol. 18, (2002), no.1, 1-26.
    4. L.Pareschi, G.Puppo, G.Russo, Central Runge-Kutta schemes for hyperbolic conservation laws, SIAM J. Sci. Comp. 26, (2005) pp.979-999. 
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