Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Exponential Peer Methods Tamer El-Azab & Rüdiger Weiner Institute of Mathematics Martin-Luther-University Halle-Wittenberg April 30, 2010 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Introduction 1 What are exponential integrators? φ -functions Computing the φ -functions Short historical overview 2 Expint Matlab package 3 Exponential Peer Methods (EPM) 4 Consistency Convergence Choosing α values Numerical Tests 5 Summary 6 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary What are exponential integrators? What are exponential integrators? Exponential integrators are those integrators which use the exponential function (and related functions) of the Jacobian or an approximation to it, inside the numerical method. Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary What are exponential integrators? What are exponential integrators? Exponential integrators are those integrators which use the exponential function (and related functions) of the Jacobian or an approximation to it, inside the numerical method. An alternative to implicit methods for the numerical solution of stiff or highly oscillatory differential equations. Many exponential integrators are designed for solving differential equations of the form y ′ ( t ) = f ( t , y ( t )) = Ty ( t )+ g ( t , y ( t )) (1) Exponential integrators have two main features: If T = 0 , then the scheme reduces to a standard scheme. 1 If g ( t , y ) = 0 for all y and t , then the scheme reproduces the 2 exact solution of (1). Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary φ -functions φ -functions The most common related functions used in exponential integrators are the so called φ -functions, which are defined as � 1 e ( 1 −θ ) z θ l − 1 φ 0 ( z ) = e z . φ l ( z ) = ( l − 1 )! d θ, l ≥ 1 , 0 The φ -functions are related by the recurrence relation φ l + 1 ( z ) = φ l ( z ) −φ l ( 0 ) φ l ( 0 ) = 1 , z l ! Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary φ -functions Computing the φ -functions The hard part of implementing exponential integrators is the evaluation of (linear combinations of) φ -functions. Some methods for evaluating the φ -function : Krylov-subspace methods. (Friesner, Tuckerman & Dornblaser 1989, Hochbruck & Lubich 1995) Leja point interpolation (Caliari & Ostermann). Using contour integrals (Schmelzer & Trefethen). RD-rational approximations (Moret & Novati 2004). Rational Krylov (Grimm & Hochbruck). Using Padè approximation combined with scaling-and-squaring. (Higham 2005) Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Outline Introduction 1 What are exponential integrators? φ -functions Computing the φ -functions Short historical overview 2 Expint Matlab package 3 Exponential Peer Methods (EPM) 4 Consistency Convergence Choosing α values Numerical Tests 5 Summary 6 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Short historical overview In 1960 Certaine, 1 Adams Moulton methods of order 2 and 3. In 1967 Lawson, 2 Generalized RK Processes (IF methods). In 1978 Friedli, 3 ETD based on explicit RK Methods of order 5. In 1998 Hochbruck and Lubich, 4 Exponential Integrators (EXP4) with inexact Jacobian. In 2003 Hochbruck and Ostermann, 5 Exponential collocation methods, convergence analysis. In 2006 Ostermann and Wright, 6 A Class of Explicit Exponential General Linear Methods. In 2009 Hochbruck, Ostermann, and Schweitzer, 7 Exponential Rosenbrock-type methods. Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Outline Introduction 1 What are exponential integrators? φ -functions Computing the φ -functions Short historical overview 2 Expint Matlab package 3 Exponential Peer Methods (EPM) 4 Consistency Convergence Choosing α values Numerical Tests 5 Summary 6 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Expint Matlab package Expint is a Matlab package designed as a tool for the numerical testing of various exponential integrators. Runge-Kutta type. Multistep type and. General Linear type. Designed by Berland, H., Skaflestad, B., and Wright, W.M. 2005. Aims of the Expint package. Create a uniform environment which enables the comparison of various integrators. Provide tools for easy visualizing of numerical behavior. Users can include problems and integrators of their own. Expint includes test problems and time stepping methods with constant step size. Computing φ -functions by using Padè approximation combined with scaling-and-squaring. Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods We will use this package for the test and comparison of Exponential Peer
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Outline Introduction 1 What are exponential integrators? φ -functions Computing the φ -functions Short historical overview 2 Expint Matlab package 3 Exponential Peer Methods (EPM) 4 Consistency Convergence Choosing α values Numerical Tests 5 Summary 6 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Exponential Peer Methods (EPM) We consider y ′ = f ( t , y ( t )) and Y mi ≈ y ( t m + c i h ) i = 1 ,..., s s-stage Peer methods. 1 All stages have the same properties. Explicit and implicit Peer methods (Podhaisky, Schmitt & Weiner 2004 – 2009). No order reduction for stiff systems observed for implicit Peer methods. Here Exponential Peer Methods. 2 Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Exponential Peer Methods (EPM) Con. s s � � Y mi = φ 0 ( α i hT ) b ij Y m − 1 , j + h A ij ( α i hT )[ f m − 1 , j − TY m − 1 , j ] (2) j = 1 j = 1 i − 1 � + h R ij ( α i hT )[ f m , j − TY m , j ] , i = 1 ,..., s . j = 1 f m − 1 , j = f ( t m − 1 + c j h , Y m − 1 , j ) . T ≈ f y for stability reasons & if T = 0 we get explicit Peer Methods. The coefficients A ij ( α i hT ) and R ij ( α i hT ) are linear combinations of the φ -functions and B = ( b ij ) s i , j = 1 ∈ R s × s . i = 1 ∈ R s is chosen to have a c = ( c i ) s i = 1 ∈ R s α = ( α i ) s and vector small number of different arguments. Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Consistency Consistency Definition 1 The exponential peer method (3) is consistent of nonstiff order p if there are constants h 0 , C > 0 such that � ∆ m , i � ≤ Ch p + 1 for all h ≤ h 0 , and for all 1 ≤ i ≤ s . The method is consistent of stiff order p , if C and h 0 may depend on ω , L g and bounds for derivatives of the exact solution, but are independent of � T � . where The nonlinear part satisfies a global Lipschitz condition � g ( t , u ) − g ( t , v ) � ≤ L g � u − v � T has a bounded logarithmic norm µ ( T ) ≤ ω. . Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary Consistency Consistency Con. Theorem 1 Consider y ′ = Ty . If the exponential Peer method satisfies the conditions s � b ij ( c j − 1 ) l = ( c i −α i ) l , l = 0 , 1 ,..., q . (3) j = 1 then it is of stiff order of consistency p = q for the linear equation y ′ = Ty . Theorem 2 Let the condition (3) be satisfied for l = 0 ,..., q . Let further s i − 1 r � � � r � � r + ( c i −α i ) r − j j ! φ j + 1 . α j + 1 � R ij c r � c j − 1 j = A ij (4) i j j = 1 j = 1 j = 0 for r = 0 ,..., q .Then the EPM is at least of stiff order of consistency p = q for (1). Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods
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