• 
A Finite Volume Method for the Relativistic Burgers Equation on a FLRW Background Spacetime
[Abstract]
[PDF]
T. Ceylan, P. G. LeFloch, and B. Okutmustur
Commun. Comput. Phys., 23 (2018), pp. 500519.
Abstract A relativistic generalization of the inviscid Burgers equation was introduced
by LeFloch and coauthors and was recently investigated numerically on a Schwarzschild
background. We extend this analysis to a FriedmannLemaýtreRobertsonWalker
(FLRW) background,which ismore challenging due to the existence of timedependent,
spatially homogeneous solutions. We present a derivation of the model of interest
and we study its basic properties, including the class of spatially homogeneous solutions.
Then, we design a secondorder accurate scheme based on the finite volume
methodology, which provides us with a tool for investigating the properties of solutions.
Computational experiments demonstrate the efficiency of the proposed scheme
for numerically capturing weak solutions.

• 
Finite volume approximation of the relativistic Burgers equation on a Schwarzschild(Anti)de Sitter spacetime
[Abstract]
[PDF]
B. Okutmustur and T. Ceylan
Turk J Math (2017) 41: 10271041.
Abstract The relativistic versions of Burgers equations on the Schwarzschild,
FLRW and de Sitter backgrounds have recently been derived and analyzed numerically via finite volume approximation
based on the concerning models. In this work, we derive the relativistic Burgers equation
on a Schwarzschild(Anti)de Sitter spacetime and introduce a secondorder Godunovtype finite volume scheme for the
approximation of discontinuous solutions to the model of interest. The effect of
the cosmological constant is also taken into account both theoretically
and numerically. The efficiency of the method for solutions containing shock and rarefaction waves are presented
by several numerical experiments.

• 
Finite Volume Method for the Relativistic Burgers Model on a (1+1)Dimensional de Sitter Spacetime
[Abstract]
[PDF]
T. Ceylan and B. Okutmustur
Math. Comput. Appl. 2016, 21(2), 16.
Abstract Several generalizations of the relativistic models of Burgers equations
have recently been established and developed on different spacetime geometries. In this work, we take
into account the de Sitter spacetime geometry, introduce our relativistic model by a technique based
on the vanishing pressure Euler equations of relativistic compressible fluids on a (1+1)dimensional
background and construct a second order Godunov type finite volume scheme to examine numerical
experiments within an analysis of the cosmological constant. Numerical results demonstrate the
efficiency of the method for solutions containing shock and rarefaction waves.

• 
Derivation of the relativistic Burgers equation on a de Sitter background
B. Okutmustur and T. Ceylan
Proccedings of 19th International Conference on Applied Mathematics, (WSEASAMATH14),
Mathematics and Computers in Science and Engineering Series 38, pp. 4147, 2014.

• 
Relativistic Burgers equations on a curved spacetime. Derivation and finite volume approximation
[Abstract]
[PDF]
P. G. LeFloch, H. Makhlof and B. Okutmustur
SIAM Journal on Numerical Analysis, Volume 50, Number 4 (2012), 21362158.
Abstract Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with
a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one
satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and
converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler
system of relativistic compressible flows on a curved background, we derive, both, the standard inviscid Burgers
equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on
curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed.
Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic
Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of
the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly,
is wellbalanced in the sense that it preserves static equilibrium solutions. Numerical experiments are presented
which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy
solutions on a curved background.

• 
Hyperbolic conservation laws on manifolds. Error estimate for finite volume schemes
[Abstract]
[PDF]
P. G. LeFloch, W. Neves and B. Okutmustur
Acta Mathematica Sinica, Volume 25, Number 7 (2009), 10411066.
Abstract Following BenArtzi and LeFloch, we consider nonlinear hyperbolic conservation laws
posed on a Riemannian manifold, and we establish an L1error estimate for a class of finite volume schemes
allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm
is of order h^1/4 at most, where h represents the maximal diameter of elements in the family of geodesic
triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch’s
theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds,
by taking into account the effects to the geometry and overcoming several new technical difficulties.
Keywords Hyperbolic conservation law, entropy solution, finite volume scheme, error estimate,
discrete entropy inequality, convergence rate.

• 
Finite Volume Schemes on Lorentzian Manifolds
[Abstract]
[PDF]
P. Amorim, P. G. LeFloch and B. Okutmustur
Communications in Mathematical Sciences, Volume 6, Number 4 (2008), 10591086.
Abstract We investigate the numerical approximation of (discontinuous) entropy solutions to
nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes
the convergence of monotone and firstorder finite volume schemes for a large class of (space and
time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy
dissipation bound, which take into account the manifold geometry and were originally discovered by
Cockburn, Coquel, and LeFloch in the (flat) Euclidian setting.
Keywords Conservation law, Lorenzian manifold, entropy condition, measurevalued solution,
finite volume scheme, convergence analysis.

• 
Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms
[Abstract]
[PDF]
P. G. LeFloch and B. Okutmustur
Far East Journal of Mathematical Sciences (FJMS), Volume 31, Issue 1 (2008), Pages 4983.
Abstract We consider nonlinear hyperbolic conservation laws, posed on a differential
(n + 1)manifold with boundary referred to as a spacetime, and
in which the “flux” is defined as a flux field of nforms depending on a
parameter (the unknown variable). We introduce a formulation of the initial
and boundary value problem which is geometric in nature and is more
natural than the vector field approach recently developed for Riemannian
manifolds. Our main assumption on the manifold and the flux field is a
global hyperbolicity condition, which provides a global timeorientation
as is standard in Lorentzian geometry and general relativity. Assuming
that the manifold admits a foliation by compact slices, we establish the
existence of a semigroup of entropy solutions. Moreover, given any two
hypersurfaces with one lying in the future of the other, we establish a
“contraction” property which compares two entropy solutions, in a (geometrically
natural) distance equivalent to the L1 distance. To carry out
the proofs, we rely on a new version of the finite volume method, which
only requires the knowledge of the given nvolume form structure on the
(n + 1)manifold and involves the total flux across faces of the elements
of the triangulations, only, rather than the product of a numerical flux
times the measure of that face.
Keywords Hyperbolic conservation law, differential manifold, flux field of forms,
entropy solution, finite volume method.

• 
Hyperbolic conservation laws on spacetimes with limited regularity
[Abstract]
[PDF]
P. G. LeFloch and B. Okutmustur
C.R. Acad. Sci. (CRAS), Paris 346 (2008), 539543.
Abstract We introduce a formulation of the initial and boundary
value problem for nonlinear hyperbolic conservation laws
posed on a differential manifold endowed with a volume form, possibly with a
boundary; in particular, this includes the important case of
Lorentzian manifolds. Only limited regularity is assumed on the geometry of
the manifold. For this problem, we establish the existence and uniqueness of
an L1 semigroup of weak solutions satisfying suitable entropy and boundary conditions.

• 
Reproducing Kernel Hilbert Spaces
[Abstract]
B. Okutmustur
Master thesis, Bilkent University, August 2005
Abstract In this thesis we make a survey of the theory of reproducing kernel Hilbert spaces
associated with positive definite kernels and we illustrate their applications for interpolation problems
of NevanlinnaPick type. Firstly we focus on the properties of reproducing kernel Hilbert spaces, generation
of new spaces and relationships between their kernels and some theorems on extensions of functions and
kernels. One of the most useful reproducing kernel Hilbert spaces, the Bergman space, is studied in details
in chapter 3. After giving a brief definition of Hardy spaces, we dedicate the last part for applications
of interpolation problems of NevanlinnaPick type with three main theorems: interpolation with a finite
number of points, interpolation with an infinite number of points and interpolation with points on the boundary.
Finally we include an Appendix that contains a brief recall of the main results from functional analysis and operator theory.
Keywords Reproducing kernel, Reproducing kernel Hilbert spaces, Bergman spaces, Hardy spaces, Interpolation, Riesz theorem.


Books : 
• 
Reproducing Kernel Hilbert Spaces: The Basics, Bergman Spaces, and Interpolation Problems
[Abstract]
with Aurelian Gheondea
LAP Lambert Academic Publishing, 2010, ISBN 9783838356310.
Abstract The theory of reproducing kernel Hilbert spaces has important applications to boundary value
problems, integral operators, harmonic and analytic functions, in conformal mappings of simply and multiplyconnected domains, in
pseudoconformal mappings, in the study of invariant Riemann metrics, in probability theory, interpolation of functions, and in many other
subjects. In this short presentation, we consider an introduction to this subject by emphasizing first the abstract theory, the Bergman kernels, and
some of their applications to interpolation of functions in the unit disc. The book is aimed to a broader audience of graduate students,
mathematicians, physicists, and engineers, and all those having an interest in getting a quick, but carefully presented, mathematically sound
basic knowledge on this domain.
Keywords Reproducing kernel, Reproducing kernel Hilbert spaces, Bergman spaces, Hardy spaces, Interpolation, Riesz theorem.

• 
Finite Volume Method For Hyperbolic Conservation Laws On Manifolds: Convergence Analysis and Error Estimation
[Abstract] Baver Okutmustur
LAP Lambert Academic Publishing, 2017, ISBN 9783330063105.
Abstract The purpose of this book is to lay out a mathematical framework for the convergence and error analysis
of the finite volume method for the discretization of hyperbolic conservation laws on manifolds. Finite Volume Method (FVM)
is a discretization approach for the numerical simulation of a wide variety physical processes described by conservation law systems.
It is extensively employed in fluid mechanics, meteorology, heat and mass transfer, electromagnetic, models of biological processes
and many other engineering applications formed by conservative systems. In this book, from one point of view, we provide a brief
description for the convergence of the FVM by approaches based on metric and differential forms. The latter can be viewed as a
generalization of the formulation and convergence of the method for general conservation laws on curved manifolds. On the other hand,
we carried over the error estimate for FVM that is established for the Euclidean setting to the curved manifolds and
obtained an expected rate of error in the L1norm.
Keywords finite volume method, conservation law, manifolds, entropy, measurevalued solution, differential forms,
convergence analysis.
