Presentation on theme: "Alexei Medovikov Tulane University"— Presentation transcript:
1 Alexei Medovikov Tulane University High order explicit methods for parabolic equations and stiff ODEs (DUMKA3, DUMKA4, ROCK2, ROCK4)Alexei MedovikovTulane UniversityA. Abdulle, A. Medovikov Second order Chebyshev methods based on orthogonal polynomials. Numerische Mathematik V90.1. pp.1-18Medovikov A.A. High order explicit methods for parabolic equations. BIT, V38,No2,ppLebedev V.I., Medovikov A.A. Method of second order accuracy with variable time steps. Izv. Vyssh. Uchebn. Zaved. Mat. no. 9, (English translation).
2 Explicit numerical methods for stiff differential equations WELCOME TO DUMKALANDExplicit numerical methods for stiff differential equations Dumka3 Examples Download DUMKA3.cpp (C++) Download DUMKA3.c (C) Download DUMKA3.f (FORTRAN) Download ROCK2/ROCK4 (rock.tar)(FORTRAN) Applications (medicine, biology, apply math ...) Motility of microorganisms Phone:(504) Address: Mathematics Department Tulane University New Orleans, LAWeb:DUMKA3 - integrates initial value problems for systems of first order ordinary differential equations y'=f(y,t). It is based on a family of explicit Runge-Kutta-Chebyshev formulas of order three. It uses optimal third order accuracy stability polynomials with the largest stability region along the negative real axis.
3 Examples of solution of stiff differential equations by explicit methods Brusselator equationNagumo nerve conduction equationBurgers equation
4 SummaryStability: Explicit methods have small stepsize , due to conditional stabilityVariable steps can be used to maximize mean stepsize of a sequence of explicit methodsOptimal sequence of explicit steps can be found in terms of roots of stability polynomials, which approximate exponential function and possess Chebyshev alternationAsymptotic formulas and orthogonal polynomials can be used to construct such polynomials, even very high degree polynomials (n > 1000)Accuracy: In order to construct high order explicit methods for non-linear ODE, we start with stability polynomials and we use B-series in order to satisfy order conditions, and build Runge-Kutta methods for non-liner ODEsEfficient stepsize control and step rejection procedure are achieved via embedded methodsFor automatic computation of spectral radius we used non-linear power method.
5 Stability analysis of explicit RK methods ODEs:Explicit Euler method:Test equation:Stability function:whereis a total stepStability region:Goal:Find stability polynomial which maximize average stepsize , given
6 Stability analysis of explicit RK methods Explicit Euler method:Stability condition of explicit Euler method:Linear stability analysis for non-linear ODEs:whereLinear stability RK methods vs. Stability RK methods?
10 Original idea of Runge-Kutta-Chebyshev methods Consider sequence of Euler steps and find an optimal polynomialas large as possibleIf we have found the optimal stability polynomial, the variable sequence ofsteps can be found in terms of the roots of the stability polynomial
11 The solution for n-stage Runge-Kutta-Chebyshev method order p=1 is given by Chebyshev stability polynomial.
13 Stability function of explicit Runge-Kutta method
14 Theorem (T. Riha): Among all polynomials of the order p the polynomial which possess Chebyshev alternant, would maximize real stability intervalor equivalently, the polynomial which possess Chebyshev alternant:has maximal possible stepsize, given stability
18 Two algorithms of computation of stability polynomials: For given n calculate weightand roots via asymptotic formula for polynomialsof the least deviation from zero:so that the polynomial satisfies (1),2. For given n calculate weightso that the polynomial satisfies (1),where is orthogonal polynomial with the weightDUMKA3,4ROCK2,4, RKC
20 Accuracy: Order conditions of Runge-Kutta methods Taylor expansions of the exact solution and numerical solution :where
21 Construction of pth order composition method Let us consider two consecutive steps by Runge-Kutta methods A and B. We call the method which is the result of one step of A and one step of B as the composition method C=B(A)Stability function of the composition method C is the product of stability functions of the methods A and BTheory of composition methods allows to calculate Taylor expansion of composition methods:
22 Given method A, define method B such that method C=B(A) will be method of the order p and stability function of the method C will be product of the stability functions of the methods A and B.Coefficients of Taylor expansion of the method B can be expressed in terms of coefficients of the methods C and B