Numerical solution of nonlinear time-dependent problems and its qualitative analysis  Page description

Help  Print 
Back »
 

Details of project

  
Identifier
67819
Type K
Principal investigator Faragó, István
Title in Hungarian Nemlineáris időfüggő feladatok numerikus megoldása és kvalitatív vizsgálata
Title in English Numerical solution of nonlinear time-dependent problems and its qualitative analysis
Keywords in Hungarian parabolikus egyenlet, nemlineáris, numerikus megoldás, kvalitatív vizsgálat
Keywords in English parabolic equation, nonlinear, qualitative analysis, numerical solution
Discipline
Mathematics (Council of Physical Sciences)90 %
Meteorology, atmospheric physics and dynamics (Council of Complex Environmental Sciences)10 %
Panel Mathematics and Computing Science
Department or equivalent Applied Analysis and Computational Mathematics (Eötvös Loránd University)
Participants Horváth, Róbert
Karátson, János
Starting date 2007-07-01
Closing date 2013-05-31
Funding (in million HUF) 9.548
FTE (full time equivalent) 4.19
state closed project
Summary in Hungarian
Parciális differenciálegyenletek a természettudomány számos jelenségének szolgálnak matematikai modelljéül (pl. elektromágnesesség, rugalmasságtan, áramlástan, reakció-diffúzió-folyamatok). E jelenségek matematikai vizsgálata és megértése egyenleteik numerikus megoldásán és kvalitatív tulajdonságaik megismerésén keresztül lehetséges. Kutatásainkban e munka két területére koncentrálunk, ezek a diszkrét maximum- és minimum-elvek, ill. iterációs módszerek a prekondicionáló operátorok elve alapján. A kvalitatív tulajdonságok kutatását a lineáris egyenletek további vizsgálata mellett (pl. időfüggő együtthatók esete) néhány nemlineáris feladatosztályra is tervezzük kiterjeszteni. Igy azok szervesen illeszkednek a prekondicionálás feladatához, mivel a nagy méret és a csatolt nemlinearitás miatt a prekondicionáló operátorok megfelelő eszköznek ígérkeznek a numerikus munkaigény csökkentéséhez.
A pályázat témája folytatása a 2003-2006 közötti „A prekondicionálás matematikai módszerei nemlineáris fizikai modellekben” OTKA pályázatnak és számos ponton kapcsolódik a résztvevők egyéb, hazai és nemzetközi együttműködéséhez. A kutatásban több PhD hallgató is részt vesz.
Summary
Partial differential equations serve as mathematical models in the description of several phenomena in the natural sciences (e.g., electromagnetism, theory of elasticity, hydrodynamics, reaction-diffusion processes).
These phenomena can be investigated and understood through their numerical simulations and the investigation of their qualitative properties. In our research we concentrate on two basic topics, namely, the discrete maximum and minimum principles, and the iterative methods based on preconditioning operators. Our aim is to continue our investigation in linear problems and extend them to nonlinear problems. These investigations fit well into the task of preconditioning, since due to the large size and the coupled nonlinearity the preconditioning operators seem to be a promising tool for the reduction of the computational work.




 

Final report

  
Results in Hungarian
1. A folytonos maximum-elv mellett megadtuk diszkrét megfelelőjét és elégséges feltételt adtunk végeselem-módszer esetére. Igazoltuk az elvet nemlineáris parciális differenciálegyenlet-rendszerekre elliptikus és időfüggő esetre . Az előjelstabilitásra feltételt adtunk szemilineáris parabolikus differenciálegyenletek véges differenciás megoldása esetére. 2. Elvégeztük az operátorszeletelés módszerének rendvizsgálatát absztrakt Cauchy-feladatra forrástagos és nemautonóm esetre is. Eredményeinket alkalmaztuk fizikai egyenletek numerikus megoldására. Alkalmaztuk a Richardson-extrapolációt a szeletelésre és a részfeladatokra alkalmazott numerikus módszer által adott konvergencia gyorsítására. Megmutattuk, hogy néhány alapvető kvalitatív tulajdonság milyen esetben öröklődik át az extrapolált kombinációra. Sikeresen alkalmaztuk az operátorszeletelés módszerét kőkopási feladatok numerikus megoldására. 3. Elliptikus feladatok iteratív megoldása keretében szuperlineárisan konvergens eljárásokat konstruáltunk és elemeztünk lineáris és nemlineáris elliptikus feladatokra, egyes konkrét modellekre és az időfüggő feladatokra. Általános leírást adtunk az ekvivalens operátorok elméletéről és alkalmazásairól végeselem- és véges differencia-módszerekre. 4. Absztrakt nemlineáris stabilitáselmélet keretében új stabilitási fogalmakat adtunk, amelyek jól illeszkednek a lineáris elmélethez.
Results in English
1. We formulated the maximum principle for parabolic problems with mixed boundary conditions and reaction and source terms. We proved sufficient conditions for the principle and the nonnegativity preservation in the finite element case . We investigated the DMP for nonlinear parabolic equations / systems and we gave sufficient conditons. Condition of the sign-stability for the mesh-parameter choice in the finite difference methods for semilinear parabolic problems was given. 2. We analyzed the convergence of the operator splitting for abstract Cauchy problems both for the nonautonomous and nonhomogeneous case. The results were applied to several physical equations. We applied Richardson extrapolation to increase the convergence order. We showed that the qualitative properties remain valid for the combination. We successfully applied the operator splitting technique to stone erosion problems. 3. For elliptic problems we constructed superlinear convergent methods and analyzed the methods for linear and nonlinear elliptic problems, for concrete tasks and also for time-dependent problems. We gave general description on the theory of equivalent operators and their application to finite difference and element methods. 4. In abstract theory we a did research in the field of nonlinear stability theory of abstract numerical analysis. We have defined new stability notions that fit well to the classical linear theory.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=67819
Decision
Yes





 

List of publications

  
I. Faragó, S. Korotov, T. Szabó: Non-negativity preservation of the discrete nonstationary heat equation in 1D and 2D, Aplimat-Journal of Applied Mathematics, 3 (2010) 60-81., 2010
I. Faragó, S. Korotov, T. Szabó: , On modifications of continuous and discrete maximum principles for reaction-diffusion problems, Adv. Appl. Math. Mech., 3 (2011) 109-120., 2011
I. Faragó, A. Havasi, Z. Zlatev: Efficient implementation of stable Richardson Extrapolation Algorithms,, Computers and Mathematics with Applications, 60 (2010) 2309–2325, 2010
I. Faragó, S. Korotov, T. Szabó: On sharpness of two-sided discrete maximum principles for reaction-diffusion problems, Aplimat-Journal of Applied Mathematics, 4 (2011) 247-254., 2011
Z. Zlatev, I. Dimov, I. Faragó at al: Richardson Extrapolated Numerical Methods for Treatment of One-Dimensional Advection Equations, Lect. Notes Comp. Sci., Springer Verlag, 6046 (2011) 198-206., 2011
I. Faragó, Zhilin Li, L. Vulkov. (editors): Finite Difference Methods: Theory and Applications, Special Issue of International Journal of Numerical Analysis & Modeling, V.3, N.2-3, 2011., 2011
I. Faragó, A. Havasi, Z. Zlatev.: Richardson extrapolation combined with the sequential splitting procedure and the θ-method,, Central European Journal of Mathematics, 2012
I. Faragó, A. Havasi, R. Horváth: , On the order of operator splitting methods for time-dependent linear systems of differential equations,, Int. J. Num. Anal. Modelling,, 2011
Z. Zlatev, A. Havasi, I. Faragó: Influence of climatic changes on pollution levels in Hungary and its surrounding countries, Atmosphere, 2011
I. Faragó, A. Havasi, R. Horváth: Numerical solution of the Maxwell equations in time-varying medium using Magnus expansion, Central European Journal of Mathematics, 2012
Z. Zlatev, I. Dimov, I. Faragó at al.: Solving advection equations by applying the Crank-Nicolson scheme combined with the Richardson Extrapolation, I. Journal of Differential Equations, 2012
I. Faragó, M. Mincsovics, I. Fekete: Notes on the basic notions in nonlinear numerical analysis, Electronic Journal of Qualitative Theory of Differential Equations, 2011
I. Faragó, J. Karátson, S. Korotov: Discrete maximum principles for nonlinear parabolic PDE systems, IMA J. Numerical Analysis, 2012
I. Faragó: Matrix maximum principles and their application, , Proc. ” The 7th Hungarian-Japanese Symposium on Discrete Mathamatics and its Applications” ,Kyoto University Press,, 2011
I. Faragó, A. Havasi, Z. Zlatev: The convergence of explicit Runge-Kutta methods combined with Richardson extrapolation, "Application of Mathematics 2012", 2012
J. Karátson: Characterizing mesh independence of Newton's method for a class of elliptic problems, SIAM J. Math. Anal., 2012
J. Karátson, B. Kovács: Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation, Comput Math Appl 65:(3), pp. 449-459, 2013
I. Faragó, F. Izsák, T. Szabó, A. Kriston: An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model, Cent. Eur. J. Math.,, 2013
I. Faragó, T. Ladics: Generalizations and error analysis of the iterative operator splitting method, Cent. Eur. J. Math., 2013
I. Faragó: Some notes on the iterative operator splitting, J. Applied and Computational Mathematics, 2013
I. Faragó, F. Izsák, T. Szabó: . An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations, Időjárás, 2013
I. Faragó: Convergence and stability constant of the theta-method, Application of Mathematics 2013, 2013
I. Faragó, Z. Zlatev, et al.: Application of Richardson Extrapolation with the Crank-Nicolson scheme for multi-dimensional advection, Application of Mathematics 2013, 2013
I. Faragó: Reliable numerical models for diffusion problems, "Supercomputer Technologies of Mathematical Modelling 2013", Editors: P. Vabisevich, V. Vasilev, Yakutsk, Russia, 2013
I. Faragó, I. Fekete: A stability approach for reaction-diffusion problems, 8th IEEE International Symposium on Applied Computational Intelligence and Informatics ", Timisoara, Romania, 2013
I. Faragó: Matrix maximum principles and their application, ” The 7th Hungarian-Japanese Symposium on Discrete Mathamatics and its Applications” Kyoto, New York, SIAM,, 2012
I. Faragó, S. Korotov, T. Szabó: On continuous and discrete maximum principles for elliptic problems with the third boundary condition, Applied Mathematics and Computation,, 2013
O. Axelsson, J. Karátson: Harmonic averages, exact difference schemes and local Green's functions in variable coefficient PDE problems, Central Eur. J. Math.11:(8) pp. 1441-1457, 2013
I. Faragó, A. Havasi, Z. Zlatev: The convergence of diagonally implicit Runge--Kutta methods combined with Richardson extrapolation, Comp. Math: Appl., 2013
J. Karátson, S. Korotov: Discrete maximum principles for FEM solutions of some nonlinear elliptic interface problems, International Journal of Numerical Analysis and Modelling, Vol. 6, No. 1, pp. 1-16, 2009
J. Karátson, T. Kurics: Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations, Numerical Analysis and Applications, Lecture Notes Comp. Sci. No. 5434, pp. 345-352, Springer, 2009
J. Karátson, S. Korotov: Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems, Appl. Math. (Prague), 54, No. 4, pp. 297-336, 2009
J. Karátson, S. Korotov: An algebraic discrete maximum principle in Hilbert space with applications to nonlinear cooperative elliptic systems, SIAM J. Numer. Anal. 47, No. 4., pp. 2518-2549., 2009
J. Karátson, T. Kurics: Some superlinear PCG methods for discretized elliptic problems, Computational Methods in Science and Engineering, Amer. Inst. Phys. Conference Proceedings, Vol. 1148, 2009; pp. 861-864, 2009
J. Karátson: Numerical Preconditioning Methods for Elliptic PDEs, in J.W. Neuberger: Sobolev Gradients and Differential Equations, 2nd Edition, Lecture Notes Math. 1670, pp. 245-258; Springer, 2010
J. Karátson: A discrete maximum principle for nonlinear elliptic systems with interface conditions, in: Large-scale Scientific Computing, Lecture Notes Comp. Sci. vol. 5910, pp. 580--587; Springer, Heidelberg, 2010
J. Karátson, S. Korotov: Discrete maximum principles for FEM solutions of nonlinear elliptic systems, in: Computational Mathematics: Theory, Methods and Applications, ed. Peter G. Chareton, Computational Mathematics and Analysis Series, NOVA Science Publishers, New York, 2010
I. Faragó, J. Karátson, S. Korotov: A discrete maximum principle for some nonlinear parabolic problems, Electr. Trans. Numer. Anal., 36 (2009-2010), pp. 149-167, 2010
I. Faragó, R. Horváth, S. Korotov: Discrete Maximum Principles for FE Solutions of Nonstationary Diffusion-Reaction Problems with Mixed Boundary Conditions, Numerical Methods of Partial Differential Equations, 27 (2011) 702-720., 2011
I. Faragó, Á. Havasi.: Operator splittings and their applications, Nova Science Publisher Inc., New York, 2009
I. Faragó, A. Havasi, Z. Zlatev: Richardson-extrapolated sequential splitting and its application, J. Comp. Appl. Math., 226, 218-227., 2009
I. Faragó: Operátorszeletelések és alkalmazásaik, Alkalmazott Matematikai Lapok, 26, 255-272., 2009
Zs. Kocsis, Z. Ferenci, I. Faragó, A. Havasi: Operator splitting in the Lagrangian air pollution transport model FLEXPART, Időjárás, Quart. J. HMS, 113, 189-202, 2009
I. Faragó: Discrete maximum principle for finite element parabolic models in higher dimensions, Math. Comp. Sim., 80, 1601–1611, 2010
A. Kriston; Gy. Inzelt, I. Faragó, T. Szabó: Simulation of Simulation of transient behavior of fuel cells by using operator splitting techniques for real time applicationsns, Computers and Chemical Engineering 34, 339–348., 2010
I. Faragó, S. Korotov, T. Szabó: Non-negativity preservation of the discrete nonstationary heat equation in 1D and 2D, Journal of Applied Mathematics, 3, 60-81., 2010
I. Faragó, A. Havasi, Z. Zlatev: Stability of the Richardson extrapolation applied together with the theta-method, Journal of Computational and Applied Mathematics, 235 (2010) 507-522., 2010
I. Faragó: Qualitative analysis of the Crank-Nicolson method for the heat conduction equation, Lect. Notes Comp. Sci., 5434, Springer Verlag, Berlin, 44-55., 2009
I. Faragó: Matrix and discrete maximum principles, Lect. Notes Comp. Sci. 5910, 563-570., 2010
Z. Zlatev, I. Farago, A. Havasi: On some stability properties of the Richardson extrapolation applied together with the theta-method, Lect. Notes Comp. Sci., Springer Verlag, 5910, 54-66., 2010
Á. Havasi, R. Horváth, Á. Nemes, T. Szabó: Investigation of a Proton Exchange Membrane Fuel Cell Model by Parameter Fitting, Fifth Conference on Finite Difference Methods: Theory and Applications (FDM'10), Rousse University press, Lozenetz, Bulgaria, 2010. június 28., 2011
O. Axelsson, J. Karátson: Condition number analysis for various forms of block matrix preconditioners, Electr. Trans. Numer. Anal. 36 (2009-2010), pp. 168-194., 2010
J. Karátson: Operator preconditioning with efficient applications for nonlinear elliptic problems, Central Eur. J. Math. 10 (1), 231-249, 2012
O. Axelsson, J. Karátson (eds.): Efficient preconditioned solution methods for elliptic partial differential equations, Bentham Science Publishers, 2011
I. Faragó, K. Georgiev, P. G. Thomsen, Z. Zlatev (Editors): Numerical Methods and Applications, Special Issue of Applied Mathematical Modelling,, Special Issue of Applied Mathematical Modelling, V.32, N.8., 2008
I. Faragó, Á. Havasi, Z. Zlatev (Editors): Advanced Numerical Algorithms for Large-Scale Computationsing, Special Issue of An International Journal Computers and Mathematics with Applications, V.55, N.10, 2008
I. Faragó, J. Karátson: Gradient--finite element method for the Saint-Venant model of elasto-plastic torsion in the hardening state, International Journal of Numerical Analysis and Modeling, 5 , 206-222., 2008
I. Antal, J. Karátson: A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems, Comput. Math. Appl. 55, 2185-2196., 2008
I. Faragó, B. Gnandt, Á. Havasi: Additive and iterative splitting methods and their numerical investigation, Computers and Mathematics with Applications, 55, 2266-2279., 2008
I. Faragó, Á. Havasi: Relationship between vanishing splitting errors and pairwise commutativity, Applied Math. Letters, 21, 10–14., 2008
I. Faragó: A modified iterated operator splitting method, Applied Mathematical Modelling, 32, 1542-1551., 2008
I. Faragó, P. Thomsen, Z. Zlatev: On the additive splitting procedures and their computer realization, Applied Mathematical Modelling, 32, 1552-1569., 2008
I. Faragó, R. Horváth: Qualitative properties of monotone linear operators, Electronic Journal of Qualitative Theory of Differential Equations, 8 , 1-15., 2008
I. Faragó, G. Inzelt, M. Kornyik, Á. Kriston, T. Szabó: Stabilization of a numerical model through the boundary conditions for the real-time simulation of fuel cells, Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering, Springer Verlag, 489-494., 2008
R. Horváth: Sufficient Conditions of the Discrete Maximum-Minimum Principle for Parabolic Problems on Rectangular Meshes, Comput Math Appl 55 (10), 2306--2317, 2008
R. Horváth: On the Sign-Stability of Numerical Solutions of One-Dimensional Parabolic Problems, Appl Math Model 32 (8), 1570-1578, 2008
J. Karátson, T. Kurics: Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems, J. Comp. Appl. Math. 212, No. 2, pp. 214-230., 2008
O. Axelsson, J. Karátson: Mesh independent convergence rates via differential operator pairs, Large-Scale Scientific Computing (LSSC'07), Lecture Notes Comp. Sci. No. 4818, Springer, pp. 3-15., 2008
J. Karátson: Superlinear PCG algorithms: symmetric part preconditioning and boundary conditions, Numer. Funct. Anal. 29, No. 5-6, pp. 1-22., 2008
J. Karátson: On the Lipschitz continuity of derivatives for some scalar nonlinearities, J. Math. Anal. Appl. 346, pp. 170-176, 2008
M. Botchev, I. Faragó, R. Horváth: Application of the operator splitting to the Maxwell equations including a source term, Appl. Num. Math., 59 , 522-541., 2009
I. Faragó, R.Horváth, S. Korotov: Discrete maximum principles parabolic problems with general boundary conditions, Journal of Applied Mathematics, 49-56., 2009
I. Faragó, P. Simon, Z. Zlatev (editors): Large Scale Scientific Computations, Special Issue of Journal of Computational and Applied Mathematics, V.226, N.2., 2009
I. Faragó: Qualitative analysis of the Crank-Nicolson method for the heat conduction equation, Lect. Notes Comp. Sci., 5434, Springer, 44-55., 2009
I. Faragó, R. Horváth: Continuous and discrete parabolic operators and their qualitative properties, IMA Numerical Analysis 29 (2009) 606-631., 2009
R. Horváth: On the Sign-Stability of Finite Difference Solutions of Semilinear Parabolic Problems, Lect. Notes Comput Sc 5434, 305-313, 2008
O. Axelsson, J. Karátson: Equivalent operator preconditioning for linear elliptic problems, Numerical Algorithms, 50, Issue 3, p. 297-380., 2009
I. Antal, J. Karátson: Mesh independent superlinear convergence of an inner-outer iterative method for semilinear elliptic interface problems, J. Comput. Appl. Math. 226, pp. 190-196., 2009



Back »