08:30
Time Integration and Spatial Discretisations (Part 1)
Chair: Francisco Gaspar
08:30
25 mins
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Approximation Properties of Discrete Boundary Value Problems for Elliptic Pseudo-Differential Equation
Vladimir B. Vasilyev
Abstract: We study a discrete variant of the following boundary value problem
\[
(Au)(x)=0,~~~x\in D,
\]
\[
(Bu)|_{\partial D}=g,
\]
where $A,B$ are simplest elliptic pseudo-differential operators acting in Sobolev--Slobodetskii spaces $H^s(D)$, $D\subset\mathbb R^m$ is a certain bounded domain, $g$ is given boundary function.
For some canonical domains, for example $D=\mathbb R^m_+=\{x\in\mathbb R^m: x=(x',x_m), x_m>0\}, D=C^a_+=\{x\in\mathbb R^m: x_m>a|x'|, a>0\}$ the solvability of discrete boundary value problems was proved for Dirichlet and Neumann boundary conditions \cite{V1,VV2,VV3}, and some comparison estimates for discrete and continuous solutions were obtained \cite{VV1}.
\begin{thebibliography}{9}
\bibitem{V1} V. Vasilyev, The periodic Cauchy kernel, the periodic Bochner kernel, and discrete pseudo-differential operators, \emph{AIP Conf. Proc.} \textbf{1863} (2017) 140014.
\bibitem{VV1} A. Vasilyev and V. Vasilyev, On a digital approximation for pseudo-differential operators, \emph{ Proc. Appl. Math. Mech.} \textbf{17} (2017) 763--764.
\bibitem{VV2}
A. Vasilyev and V. Vasilyev, Pseudo-differential operators and equations in a discrete half-space,
\emph{Math. Model. Anal.}
\textbf{23} (2018), 492--506.
\bibitem{VV3} A. Vasilyev and V. Vasilyev, On some discrete potential like operators, \emph{Tatra Mt. Math. Publ.} \textbf{71} (2018), 195--212.
\end{thebibliography}
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08:55
25 mins
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Godunov Type Solvers for Euler Systems with Friction Terms
Aekta Aggarwal, Ganesh Vaidya, G,D, Veerappa Gowda
Abstract: The paper deals with the construction of the numerical schemes for the one--dimensional Euler system with the Coulomb-like friction term with a constant friction coefficient
ρ_t+(ρu)_x=0,(ρu)_t+(ρu^2- εA/ρ)_x=βρ (0.1 )
Where ρ and u denote the gas density and velocity respectively and ε,A>0. This system known as, Chaplygin gas equation, is a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. Since it possesses a negative pressure term, it allows mass concentration and can be used to depict dark energy models in cosmology. Various analytical and numerical studies have been done to study the formation of delta shock waves for the homogeneous system with β=0.
In this study, we are interested in the case β≠0 which depicts damping, friction and relaxation effect. As ε→0, the system converges to the so-called pressureless gas dynamics model
ρ_t+(ρu)_x=0,(ρu)_t+(ρu^2 )_x=βρ (0.2)
with body force as a source and is used to describe the motion process of free particles sticking under collision in the low temperature and the information of large-scale structures in the universe. The Riemann
solutions consisting of the delta shock wave and vacuum were derived in [2].
In this study, we are interested in forming Godunov-type Solvers for the (0.1) and (0.2) based on discontinuous flux were introduced in [1] for non--linear fluxes. The paper [1] is limited to non-linear fluxes only and this paper extends this idea to the equations of the type
ρ_t+(a(x)ρ)_x=0 (0.3)
with a(x) being discontinuous. The homogeneous version of system (0.2) consists of 2 equations of the type (0.3) and the new Godunov solver for (0.3) is used for the homogeneous system (0.2) equation by equation. For the non homogeneous version, the source term is incorporated in the system through a suitable change of variables so that both the equations of the (0.2) look like (0.3) with same coefficient a(x). The stability properties like positivity of ρ and boundedness of u are established using monotonicity of the flux. Numerical experiments are conducted and results are compared with other existing results. The scheme is shown to perform better than previous studies, both in location and strength δ- shock.
References
[1] A. Adimurthi, J. Jaffr´e, and G. D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal., 42 (2004), pp. 179–208.
[2] C. Shen, The riemann problem for the pressureless euler system with the coulomb-like friction term, IMA Journal of applied Mathematics, 81 (2015), pp. 76–99.
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09:20
25 mins
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Towards a simple and high-order numerical method for PDEs on evolving domains
Yimin Lou, Christoph Lehrenfeld
Abstract: Mathematical models in physics, chemistry, biology and engineering often involve partial differential equations (PDEs) posed on moving domains. In order to handle the physical domains that may exhibit strong deformations or even topology changes, the geometrically unfitted finite element methods have been developed to find a numerical solution of such a problem. To this end, we represent the physical domains, embedded and evolving smoothly in a time-independent background, implicitly by level set functions. However, the numerical methods usually face a challenge of providing a solution of both accuracy and efficiency. A new Eulerian finite element method proposed by Lehrenfeld and Olshanskii uses finite difference (FD) discretization in time and an addtional stabilization term in the spatial domains, which is provably convergent in space and time. While this has been proved only for low-order approximation in time, we advance the method to provably high-order accuracy in both space and time. Our method is based on a spatially stabilized high-order FD-type discretization of time derivatives, coupled with an isoparametric geometrically unfitted finite element discretization in space. The level set domains are approximated to high order of accuracy by the technique of isoparametric mappings due to Lehrenfeld. With the high-order approximation of the geometry, some high-order FD-type time discretizations can be combined to describe the motion of the physical domains, e.g., the family of BDF schemes, in order to achieve a solution of uniformly higher order of accuracy.
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09:45
25 mins
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A shock capturing scheme for non-convex special relativistic hydrodynamics
Susana Serna, Antonio Marquina, Jose Maria Ibanez
Abstract: The equations of special relativistic hydrodynamics (SRHD) form a nonlinear system of
conservation laws which is closed with the constitutive relations represented by an equation of
state (EOS) characterizing the equilibrium thermodynamic properties of the considered material.
The thermodynamics, through the EOS, provides the classical or non-classical (convex
or non-convex) character of the wave structure.
There are many astrophysical scenarios governed by relativistic (magneto-) hydrodynamical
processes. A very outstanding example is the recent detection of GW170817, the gravitational
wave signal coming from two colliding neutron stars ([1]). The very rich and complex
thermodynamics involved in such a system is still an open problem for the community of
nuclear physicists interested in the properties of dense matter. A necessary (not sufficientt)
condition for a non-convex thermodynamics is the non-monotonicity of the local speed of
sound.
The numerical approximation of the solution of the SRHD system of equations is a challenging
task. The ultrarelativistic regime is known because of the extremely strong shock
structures appearing in the dynamics. The equations are strongly coupled through the Lorentz
factor, and the relativistic enthalpy, bringing up a very nonlinear wave interaction [2,3].
Moreover conserved and primitive variables are related through a set of non-linear equations
that need to be solved in each point at every time step.
Traditionally, numerical schemes have been designed relying on the classical behavior of
the wave structure (convexity of the system of equations) taking the ideal gas EOS as the
model EOS. In this study we deal with the numerical approximation of the complex structure
of SRHD when the system is closed with a non-convex EOS. We consider a recently introduced
phenomenological EOS ([4]) that mimics the loss of classical behavior when the
fluid enters into a non-convex - thermodynamically - region in the relativistic regime. We design a
flux formulation approach to approximate the solution of Riemann problems in SRHD such that the
non-classical dynamics is detected and well resolved. We also propose an iterative procedure
to recover primitive variables ensuring convergence around the boundary of the non-convex
region.
References
[1] LIGO Scientic Collaboration and Virgo Collaboration, GW170817: Observation of Gravitational
Waves from a Binary Neutron Star Inspiral, Phys.Rev.Letters, vol. 119, 161101,
2017.
[2] LeVeque RJ, Mihalas, D, Dor, EA, Muller, E, Computational methods for astrophysical
fluid flow, Saas-Fee Advanced Courses, 27, Springer-Verlag, New York, 1998.
[3] Mart, J. M. and Muller, E., Numerical Hydrodynamics in Special Relativity, Living Rev.
Relativ. 6: 7. https://doi.org/10.12942/lrr-2003-7, 2003.
[4] Iba~nez, J.M., Marquina, A., Serna, S., Aloy, M.A., Anomalous dynamics triggered by a
non-convex equation of state in relativistic
ows, Monthly Notices of the Royal Astronomical
Society, 476, 1100, 2018.
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