THE UNIVERSITY OF BIRMINGHAM

THE SCHOOL OF MATHEMATICS

THE APPLIED MATHEMATICS GROUP

N.B.PETROVSKAYA

Coarse grids

Higher order schemes

Numerical integration

Multiple solutions

References

My research interests mainly concern applied numerical analysis and computer simulation of complex physical and engineering problems. Some topics of my current research are enlisted below. You may also wish to learn more about my research from the description of my current   graduate and postgraduate projects .


The coarse grid problem


Function approximation on coarse grids is a challenging topic, as asymptotic error estimates cannot be applied on a coarse mesh. The problem consists of two issues equally important for practical applications. The first question is how to obtain the required result (e.g., a function interpolant or an integral estimate) with the maximum accuracy possible on a given grid with a fixed number of grid nodes. The second issue, which is directly related to the first one, is to evaluate how reliable the result is when the data array has a small dimension [1].

The coarse mesh problem is a very general computational problem that has a number of applications. In particular, I deal with problems that come from mathematical ecology [2] and computational aerodynamics [3], [4].

Higher order discretization schemes


Higher order discretization schemes are one of the most important topics of the research in modern computational fluid dynamics. A discontinuous Galerkin (DG) discretization is a higher order scheme that combines advantages of finite element and finite volume approaches. The various issues related to implementation of DG schemes in computational aerodynamics problems have been investigated in [5], [6], [7].

The hyperbolic systems of conservation laws present a wide class of problems where the DG method can be successfully applied. However, there are still a number of open questions when steady state solutions to conservation laws are considered. In particular, a numerical solution may exhibit strong oscillations over the entire domain of computation, if a high order DG discretization is employed to solve a steady state problem. Limiters should then be incorporated into the scheme to eliminate spurious solution oscillations. Limiters for high order DG schemes have been studied in [8].

Numerical integration in ecological problems


Numerical integration of sampled data arises in many ecological problems, where the integrand function is available from experimental measurements only. One important field of research is the problem of biological invasion where an accurate evaluation of the population size from the spatial density distribution is required for a given biological species. The study of biological invasion has many applications and I am interested in a particular problem of pest insect monitoring, as an accurate estimate of the total number of pest insects is crucial for making a reliable decision about the use of of pesticides. [9].

The data for evaluation of the pest population size are usually collected by trapping. The main difficulty associated with the estimation of the total number of pest insects from trap counts is that the number of traps cannot be made large enough to ensure that the integral estimate is accurate and new numerical techniques are required to evaluate the integral [1], [10]. My current research is to design suitable methods of numerical integration (and to compare them with statistical methods where possible) for the following problems:

1. Accurate estimation of pest population abundance when the data are sparse.

2. Accurate estimation of pest population abundance when the data are noisy.

3. Accurate estimation of pest population abundance when the data are collected in a domain with complex spatial geometry.

Multiple solutions to nonlinear problems


It is very often that a solution to a nonlinear problem can be non-unique. The non-uniqueness of a nonlinear solution is a challenging task from a computational viewpoint. In many cases new computational algorithms have to be developed for the numerical solution, as solution bifurcations arising due to nonlinearity of the problem result in divergency of standard numerical methods [11].

In numerical solution of nonlinear steady state problems, Newton's method is an approach suggesting potential computational benefits in comparison with time dependent algorithms. However, a drawback of this numerical strategy is that a transient solution in the Newton method may experience jumps from one solution branch to another, until the basin of attraction is approached. Those local bifurcations are dangerous as they result in a divergent solution. The solution bifurcations must be controlled by a numerical algorithm in order to converge to a physically relevant branch of the solution [12].

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References


[1]  N.B.Petrovskaya, N.L.Embleton.   Evaluation of Peak Functions on Ultra-Coarse Grids.   Proc. R. Soc. A, 2013, vol.469 (article in press) doi: 10.1098/rspa.2012.0665

[2]  N.B.Petrovskaya, S. V. Petrovskii.   The Coarse-Grid Problem in Ecological Monitoring.   Proc. R. Soc. A, 2010, vol.466, pp. 2933-2953, doi: 10.1098/rspa.2010.0023

[3]  N.B.Petrovskaya.   Data Dependent Weights in Discontinuous Weighted Least-Squares Approximation with Anisotropic Support.   Calcolo, 2011, vol.48(1), pp.127-143, doi: 10.1007/s10092-010-0032-7

[4]   N.B.Petrovskaya.   Discontinuous Weighted Least-Squares Approximation on Irregular Grids.   CMES: Computer Modeling in Engineering & Sciences, 2008, vol.32(2), pp.69-84.

[5]   A. V. Wolkov, Ch. Hirsch, N.B.Petrovskaya.   Application of a Higher Order Discontinuous Galerkin Method in Computational Aerodynamics.   Mathematical Modeling of Natural Phenomena, 2011, vol.6(3), pp.245-272 (invited paper).

[6]   A.V.Wolkov, N.B.Petrovskaya.   Higher Order Discontinuous Galerkin Method for Acoustic Pulse problem.   Comput. Phys. Commun., 2010, vol.181, pp. 11861194.

[7]   N.B.Petrovskaya, A.V.Wolkov, S.V.Lyapunov.   Modification of Basis Functions in High Order Discontinuous Galerkin Schemes for Advection Equation. Appl. Math. Model., 2008, vol.32(5), pp.826-835.

[8]   N.B.Petrovskaya.   Two Types of Solution Overshoots in Discontinuous Galerkin Discretization Schemes.   Comm. Math. Sci., 2005, vol.3(2), pp.235-249.

[9]   N.B.Petrovskaya, E.Venturino.   Numerical Integration of Sparsely Sampled Data.   Simulat. Modell. Pract. Theory, 2011, vol.19(9), pp.18601872, doi:10.1016/j.simpat.2011.05.003

[10]   N.B.Petrovskaya, S.V.Petrovskii, A.K.Murchie.   Challenges of Ecological Monitoring: Estimating Population Abundance From Sparse Trap Counts.   J.R.Soc.Interface, 2011, doi:10.1098/rsif.2011.0386

[11]   K.V.Brushlinsky, M.S.Mikhaylova, N.B.Petrovskaya, N.M.Zueva.   On Uniqueness and Stability of Solutions to 2-D Plasmastatics Problems.  J. Math. Mod., 1995, vol.7(4), p.73-86.

[12]   N.B.Petrovskaya.   On Oscillations in Discontinuous Galerkin Discretization Schemes for Steady State Problems.   SIAM J.Sci.Comput, 2006, vol.27(4), p. 1329 -1346.