Casa seminar Rosenbrock (AMF) methods November 12 2008, Marly van - PowerPoint PPT Presentation

Casa seminar Rosenbrock (AMF) methods November 12 2008, Marly van Geel Where innovation starts

Contents 2/25 Introduction 3 Runge-Kutta methods 4 Rosenbrock Methods 6 Model problem 11 Numerical results 13 Conclusion 24 / department of mathematics and computer science November 12th 2008

Introduction 3/25 u t + ( au ) x = ( du x ) x + f ( u ) , / department of mathematics and computer science November 12th 2008

Runge-Kutta methods 4/25 Runge-Kutta methods: We have dw dt = F ( w ) .  s �  w n +1 = w n + b i k i     i =1 � � s �  k i = τF w n + α ij k j i = 1 , . . . , s     j =1 Explicit if α ij = 0 for j ≥ i . Otherwise implicit. / department of mathematics and computer science November 12th 2008

Diagonal Implicit Runge-Kutta 5/25 This method is  s �  w n +1 = w n + b i k i     i =1 � � i �  k i = τF w n + α ij k j i = 1 , . . . , s     j =1 Newton iteration to calculate k i / department of mathematics and computer science November 12th 2008

Rosenbrock Methods 6/25 Rosenbrock methods are Runge-Kutta type methods. s  �  w n +1 = w n + b i k i     i =1 � � i − 1 i � �  k i = τF w n + α ij k j + τJ γ ij k j i = 1 , . . . , s     j =1 j =1 With J = F ′ ( w n ) . Jacobian matrix in the integration formula. / department of mathematics and computer science November 12th 2008

Rosenbrock Methods (II) 7/25 A-stability or L-stability order p order conditions 1 b 1 + b 2 + b 3 = 1 b 2 d 2 + b 3 d 3 + b 4 d 4 = 1 2 − γ 2 b 2 c 2 2 + b 3 c 2 3 + b 4 c 2 4 = 1 3 3 b 3 β 32 d 2 + b 4 ( β 42 d 2 + β 43 d 3 ) = 1 6 − γ + γ 2 where i − 1 i − 1 � � β ij = α ij + γ ij , c i = α ij , d i = β ij . j =1 j =1 / department of mathematics and computer science November 12th 2008

Rosenbrock Methods (III) 8/25 The 1-stage method w n +1 = w n + k 1 , k 1 = τF ( w n ) + γτJk 1 . Stability function R ( z ) = 1 + (1 − γ ) z 1 − γz Order 2 if γ = 1 2 . A-stability if γ ≥ 1 2 and L-stability if γ = 1 . / department of mathematics and computer science November 12th 2008

Rosenbrock Methods (IV) 9/25 The 2-stage method We have w n +1 = w n + b 1 k 1 + b 2 k 2 , k 1 = τF ( w n ) + γτJk 1 k 2 = τF ( w n + α 21 k 1 ) + γ 21 τJk 1 + γτJk 2 with α 21 = 1 γ 21 = − γ b 1 = 1 − b 2 , , 2 b 2 b 2 Stability function R ( z ) = 1 + (1 − 2 γ ) z + ( 1 2 − 2 γ + γ 2 ) z 2 (1 − γz ) 2 √ A-stability if γ ≥ 1 4 and L-stability if γ = 1 ± 1 2 . 2 / department of mathematics and computer science November 12th 2008

Rosenbrock AMF methods 10/25 Splitting based on special approximations of the Jacobian matrix. w ′ ( t ) = F ( w ( t )) = F 1 ( w ( t )) + · · · + F s ( w ( t )) One-stage method k i = τ ( I − γτJ ) − 1 F ( w n ) w n +1 = w n + k i , w n +1 = w n + τB − 1 F ( w n ) , B = I − γτJ, J = F ′ ( w n ) . Replace B = I − γτJ by ˜ B = ( I − γτJ 1 )( I − γτJ 2 ) · · · ( I − γτJ s ) / department of mathematics and computer science November 12th 2008

Model problem 11/25 Model problem u t + au x = εu xx + λu (1 − u ) , 0 < x < 1 , t > 0 u x (0 , t ) = 0 , t > 0 u (1 , t ) = (1 + sin( wt ) / 2 , t > 0 u ( x, 0) = v ( x ) , 0 < x < 1 . Write problem as ordinary differential equation. / department of mathematics and computer science November 12th 2008

Model problem 12/25 The ODE becomes     − 2 2 0 0 1 − 2 1 − 1 0 1     ε u − a ... ... ... ... ... ...     u t = u     h 2 2 h     1 − 2 1 − 1 0 1     1 − 2 − 1 0   0 0 � ε   � 1 + sin( wt ) � h 2 − a � .  .  + + λ (1 − u j ) u j .   2 h 2   0   1 = ( A d − A a ) u + b + f = F ( u ) Now apply Rosenbrock (AMF) methods. / department of mathematics and computer science November 12th 2008

Numerical results 13/25 Rosenbrock methods The 1-stage method w n +1 = w n + k 1 , k 1 = τF ( w n ) + γτJk 1 . J is the Jacobian matrix F ′ ( w n ) . a = − 1 , ε = 0 . 01 , λ = 1 , frequency = 10 , N x = 100 , N t = 100 . Unstable γ = 0 . 1 Unstable γ = 0 . 2 . / department of mathematics and computer science November 12th 2008

Numerical results 14/25 Stable γ = 0 . 4 Stable γ = 0 . 6 . Stable γ = 1 . 2 Stable γ = 2 . 0 . / department of mathematics and computer science November 12th 2008

Numerical results 15/25 a = − 1 , ε = 1 , λ = 1 , frequency = 10 , N x = 100 , N t = 100 . Unstable γ = 0 . 4 Stable γ = 0 . 5 . / department of mathematics and computer science November 12th 2008

Numerical results 16/25 The 2-stage method w n +1 = w n + b 1 k 1 + b 2 k 2 , k 1 = τF ( w n ) + γτJk 1 k 2 = τF ( w n + α 21 k 1 ) + γ 21 τJk 1 + γτJk 2 with α 21 = 1 γ 21 = − γ b 1 = 1 − b 2 , , 2 b 2 b 2 / department of mathematics and computer science November 12th 2008

Numerical results 17/25 a = − 1 , ε = 0 . 01 , λ = 1 , frequency = 10 , b 2 = 0 . 4 , N x = 100 , N t = 100 . Unstable γ = 0 . 1 Stable γ = 0 . 2 . / department of mathematics and computer science November 12th 2008

Numerical results 18/25 Stable γ = 0 . 3 Stable γ = 0 . 5 . Stable γ = 1 . 0 Stable γ = 1 . 5 . / department of mathematics and computer science November 12th 2008

Numerical results 19/25 a = − 1 , ε = 1 , λ = 1 , frequency = 10 , b 2 = 0 . 4 , N x = 100 , N t = 100 . Unstable γ = 0 . 2 Stable γ = 0 . 3 . / department of mathematics and computer science November 12th 2008

Numerical results 20/25 Rosenbrock AMF methods One-stage method w n +1 = w n + τB − 1 F ( w n ) , B = I − γτJ, J = F ′ ( w n ) . Replace B = I − γτJ by ˜ B = ( I − γτJ 1 )( I − γτJ 2 ) · · · ( I − γτJ s ) Approximation is B = ( I − γτA )( I − γτf ′ ) = B + ( γτ ) 2 Af ′ = B + O ( τ 2 ) ˜ / department of mathematics and computer science November 12th 2008

Numerical results 21/25 a = − 1 , ε = 0 . 01 , λ = 1 , frequency = 10 , b 2 = 0 . 4 , N x = 100 , N t = 100 . Unstable γ = 0 . 1 Unstable γ = 0 . 2 . / department of mathematics and computer science November 12th 2008

Numerical results 22/25 Stable γ = 0 . 3 Stable γ = 0 . 4 . Stable γ = 0 . 6 Stable γ = 1 . 2 . / department of mathematics and computer science November 12th 2008

Numerical results 23/25 a = − 1 , ε = 1 , λ = 1 , frequency = 10 , b 2 = 0 . 4 , N x = 100 , N t = 100 . ε = 1 , Unstable γ = 0 . 4 Stable γ = 0 . 5 . / department of mathematics and computer science November 12th 2008

Conclusion 24/25 Rosenbrock 1-stage method A-Stability when γ ≥ 1 2 . Rosenbrock 2-stage method A-Stability when γ ≥ 1 4 . Rosenbrock AMF method A-Stability when γ ≥ 1 2 . / department of mathematics and computer science November 12th 2008

Questions 25/25 Questions? / department of mathematics and computer science November 12th 2008