A new approach to control the global error of numerical methods for - PowerPoint PPT Presentation

Double Quasi-Consistency HISTORY on Quasi-Consistent Integration: Skeel discovered the property of quasi-consistency in 1976. Skeel and Jackson found the first quasi-consistent methods among fixed-stepsize Nordsieck formulas in 1977. ▽ A new approach to control the global error of numerical methods for differential equations – p.8/59

Double Quasi-Consistency HISTORY on Quasi-Consistent Integration: Skeel discovered the property of quasi-consistency in 1976. Skeel and Jackson found the first quasi-consistent methods among fixed-stepsize Nordsieck formulas in 1977. Kulikov and Shindin proved in 2006 that conventional Nordsieck formulas cannot exhibit the quasi-consistent behaviour on variable meshes because of the order reduction phenomenon. A new approach to control the global error of numerical methods for differential equations – p.8/59

Double Quasi-Consistency HISTORY on Quasi-Consistent Integration (cont.): ▽ A new approach to control the global error of numerical methods for differential equations – p.9/59

Double Quasi-Consistency HISTORY on Quasi-Consistent Integration (cont.): In 2009, Weiner et al. constructed actual variable-stepsize quasi-consistent numerical schemes in the family of explicit two-step peer formulas. ▽ A new approach to control the global error of numerical methods for differential equations – p.9/59

Double Quasi-Consistency HISTORY on Quasi-Consistent Integration (cont.): In 2009, Weiner et al. constructed actual variable-stepsize quasi-consistent numerical schemes in the family of explicit two-step peer formulas. Kulikov proved in the same year that there exists no doubly quasi-consistent Nordsieck formula. A new approach to control the global error of numerical methods for differential equations – p.9/59

Double Quasi-Consistency Thus, the first issue is: Existence of Doubly Quasi-Consistent Numerical Schemes ▽ A new approach to control the global error of numerical methods for differential equations – p.10/59

Double Quasi-Consistency Thus, the first issue is: Existence of Doubly Quasi-Consistent Numerical Schemes Further, we prove Existence of Doubly Quasi-Consistent Numerical Schemes in the family of fixed-stepsize s -stage Explicit Parallel Peer methods (EPP-methods) A new approach to control the global error of numerical methods for differential equations – p.10/59

Fixed-Stepsize EPP Methods We deal further with numerical schemes of the form s s � � x ki = b ij x k − 1 ,j + τ a ij g ( t k − 1 ,j , x k − 1 ,j ) , (3) j =1 j =1 i = 1 , 2 , . . . , s, ▽ A new approach to control the global error of numerical methods for differential equations – p.11/59

Fixed-Stepsize EPP Methods We deal further with numerical schemes of the form s s � � x ki = b ij x k − 1 ,j + τ a ij g ( t k − 1 ,j , x k − 1 ,j ) , (3) j =1 j =1 i = 1 , 2 , . . . , s, or in the matrix form X k = ( B ⊗ I m ) X k − 1 + τ ( A ⊗ I m ) g ( T k − 1 , X k − 1 ) where T k := ( t ki ) s i =1 , X k := ( x ki ) s i =1 , g ( T k , X k ) := g ( t ki , x ki ) s i =1 , � s � s � � A := a ij i,j =1 , B := b ij i,j =1 . A new approach to control the global error of numerical methods for differential equations – p.11/59

Fixed-Stepsize EPP Methods We deal further with numerical schemes of the form s s � � x ki = b ij x k − 1 ,j + τ a ij g ( t k − 1 ,j , x k − 1 ,j ) , (3) j =1 j =1 i = 1 , 2 , . . . , s, or in the matrix form X k = ( B ⊗ I m ) X k − 1 + τ ( A ⊗ I m ) g ( T k − 1 , X k − 1 ) where T k := ( t ki ) s i =1 , X k := ( x ki ) s i =1 , g ( T k , X k ) := g ( t ki , x ki ) s i =1 , � s � s � � A := a ij i,j =1 , B := b ij i,j =1 . A new approach to control the global error of numerical methods for differential equations – p.12/59

Fixed-Stepsize EPP Methods DEFINITION 1: The peer method (3) is consistent of order p if and only if the following order conditions hold: s b ij ( c j − 1) l + l a ij ( c j − 1) l − 1 � � AB i ( l ) := c l � i − = 0 , l ≤ p. j =1 ▽ A new approach to control the global error of numerical methods for differential equations – p.13/59

Fixed-Stepsize EPP Methods DEFINITION 1: The peer method (3) is consistent of order p if and only if the following order conditions hold: s b ij ( c j − 1) l + l a ij ( c j − 1) l − 1 � � AB i ( l ) := c l � i − = 0 , l ≤ p. j =1 THEOREM 1: The peer method (3) of order p is doubly quasi-consistent if and only if its coefficients a ij , b ij and c i satisfy the following conditions: AB ( l ) = 0 , l = 0 , 1 , . . . , p − 1 , B · AB ( p ) = 0 , B · AB ( p + 1) = 0 , A · AB ( p ) = 0 . A new approach to control the global error of numerical methods for differential equations – p.13/59

Fixed-Stepsize EPP Methods With the use of Theorem 1, we yield the following doubly quasi-consistent EPP-method (3) presented by its coefficients:   89 23 − 5 144 48 36   − 133 29 55 A =  ,   144 48 36  − 37 41 10 144 48 9     11 1 − 1 1 18 2 9 4     11 1 − 1 1 B =  , c =  .     18 2 9 2   11 1 − 1 1 18 2 9 ▽ A new approach to control the global error of numerical methods for differential equations – p.14/59

Fixed-Stepsize EPP Methods With the use of Theorem 1, we yield the following doubly quasi-consistent EPP-method (3) presented by its coefficients:   89 23 − 5 144 48 36   − 133 29 55 A =  ,   144 48 36  − 37 41 10 144 48 9     11 1 − 1 1 18 2 9 4     11 1 − 1 1 B =  , c =  .     18 2 9 2   11 1 − 1 1 18 2 9 This is a 3-stage explicit parallel peer method of order 2. A new approach to control the global error of numerical methods for differential equations – p.14/59

Global Error Estimation & Control TRUE ERROR EVALUATION: Having used an embedded peer method (3) with coefficients A emb , B emb and c , we arrive at the error evaluation scheme of the form � � ∆ 1 X k = ( B emb − B ) ⊗ I m X k − 1 (4) � � + τ ( A emb − A ) ⊗ I m g ( T k − 1 , X k − 1 ) where ∆ 1 X k denotes the principal term of the true error of the doubly quasi-consistent peer method and X k − 1 implies the numerical solution computed by the same peer method. The global error estimation formula (4) is cheap. A new approach to control the global error of numerical methods for differential equations – p.15/59

Global Error Estimation & Control THEOREM 2: Let the peer method (3) be doubly quasi-consistent and of order p . Then formula (4) computes the principal term of its true error at grid points if and only if the coefficients A emb , B emb and c of the embedded peer method satisfy the following conditions: AB ( l ) emb = 0 , l = 0 , 1 , . . . , p, B emb · AB ( p ) = 0 where the vectors AB ( l ) emb , l = 0 , 1 , . . . , p , are calculated for the coefficients of the embedded formula (3) and the vector AB ( p ) is evaluated for the coefficients of the doubly quasi-consistent peer method in the embedded pair. A new approach to control the global error of numerical methods for differential equations – p.16/59

Global Error Estimation & Control With the use of Theorem 2, the embedded peer method (3) for the doubly quasi-consistent peer scheme above is chosen to have the coefficients:   − 1 47 151 18 96 288   7 − 35 341 A emb =  ,   18 96 288  58 − 476 1069 18 96 288     11 1 − 1 1 18 2 9 4     11 1 − 1 1 B emb =  , c =  .     18 2 9 2   11 1 − 1 1 18 2 9 ▽ A new approach to control the global error of numerical methods for differential equations – p.17/59

Global Error Estimation & Control With the use of Theorem 2, the embedded peer method (3) for the doubly quasi-consistent peer scheme above is chosen to have the coefficients:   − 1 47 151 18 96 288   7 − 35 341 A emb =  ,   18 96 288  58 − 476 1069 18 96 288     11 1 − 1 1 18 2 9 4     11 1 − 1 1 B emb =  , c =  .     18 2 9 2   11 1 − 1 1 18 2 9 This embedded formula is of classical order 2 and has the local error of O ( τ 3 ) . A new approach to control the global error of numerical methods for differential equations – p.17/59

Global Error Estimation & Control GLOBAL ERROR CONTROL ALGORITHM: 1. k := 0 , τ := τ int ; ( τ int , γ ∈ (0 , 1) are set); 2. While t k < t end do, t k +1 := t k + τ , compute X k +1 , ∆ 1 X k +1 ; 3. If max � ∆ 1 X k +1 � > ǫ g , k � 1 /p � � ∆ 1 ˜ then τ := γτ ǫ g / max X k +1 � , go to 1, k else Stop. A new approach to control the global error of numerical methods for differential equations – p.18/59

Global Error Estimation & Control TEST PROBLEM 1: Simple Problem � �� 1 ( t ) = 2 tx 1 / 5 x ′ 2 ( t ) x 4 ( t ) , x ′ � 2 ( t ) = 10 t exp 5 x 3 ( t ) − 1 x 4 ( t ) , x ′ x ′ � � 3 ( t ) = 2 tx 4 ( t ) , 4 ( t ) = − 2 t ln x 1 ( t ) , where t ∈ [0 , 3] , x (0) = (1 , 1 , 1 , 1) T . ▽ A new approach to control the global error of numerical methods for differential equations – p.19/59

Global Error Estimation & Control TEST PROBLEM 1: Simple Problem � �� 1 ( t ) = 2 tx 1 / 5 x ′ 2 ( t ) x 4 ( t ) , x ′ � 2 ( t ) = 10 t exp 5 x 3 ( t ) − 1 x 4 ( t ) , x ′ x ′ � � 3 ( t ) = 2 tx 4 ( t ) , 4 ( t ) = − 2 t ln x 1 ( t ) , where t ∈ [0 , 3] , x (0) = (1 , 1 , 1 , 1) T . The exact solution is well-known: sin t 2 � 5 sin t 2 � � � x 1 ( t ) = exp , x 2 ( t ) = exp , x 3 ( t ) = sin t 2 + 1 , x 4 ( t ) = cos t 2 . A new approach to control the global error of numerical methods for differential equations – p.19/59

Global Error Estimation & Control TEST PROBLEM 2: Restricted Three Body Problem x 1 ( t ) + µ 2 x 1 ( t ) − µ 1 x ′′ 1 ( t ) = x 1 ( t ) + 2 x ′ 2 ( t ) − µ 1 − µ 2 , y 1 ( t ) y 2 ( t ) x 2 ( t ) x 2 ( t ) x ′′ 2 ( t ) = x 2 ( t ) − 2 x ′ 1 ( t ) − µ 1 y 1 ( t ) − µ 2 y 2 ( t ) , � 3 / 2 � 3 / 2 � ( x 1 ( t )+ µ 2 ) 2 + x 2 � ( x 1 ( t ) − µ 1 ) 2 + x 2 y 1 ( t )= 2 ( t ) , y 2 ( t )= 2 ( t ) where t ∈ [0 , T ] , T = 17 . 065216560157962558891 , µ 1 = 1 − µ 2 and µ 2 = 0 . 012277471 . The initial values are: x 1 (0) = 0 . 994 , x ′ 1 (0) = 0 , x 2 (0) = 0 , x ′ 2 (0) = − 2 . 00158510637908252240 . ▽ A new approach to control the global error of numerical methods for differential equations – p.20/59

Global Error Estimation & Control TEST PROBLEM 2: Restricted Three Body Problem x 1 ( t ) + µ 2 x 1 ( t ) − µ 1 x ′′ 1 ( t ) = x 1 ( t ) + 2 x ′ 2 ( t ) − µ 1 − µ 2 , y 1 ( t ) y 2 ( t ) x 2 ( t ) x 2 ( t ) x ′′ 2 ( t ) = x 2 ( t ) − 2 x ′ 1 ( t ) − µ 1 y 1 ( t ) − µ 2 y 2 ( t ) , � 3 / 2 � 3 / 2 � ( x 1 ( t )+ µ 2 ) 2 + x 2 � ( x 1 ( t ) − µ 1 ) 2 + x 2 y 1 ( t )= 2 ( t ) , y 2 ( t )= 2 ( t ) where t ∈ [0 , T ] , T = 17 . 065216560157962558891 , µ 1 = 1 − µ 2 and µ 2 = 0 . 012277471 . The initial values are: x 1 (0) = 0 . 994 , x ′ 1 (0) = 0 , x 2 (0) = 0 , x ′ 2 (0) = − 2 . 00158510637908252240 . Its solution-path is periodic. A new approach to control the global error of numerical methods for differential equations – p.20/59

Global Error Estimation & Control NUMERICAL RESULTS for our Test Problems: Accuracy Graph Accuracy Graph 0 0 10 10 Exact Error Exact Error Error Estimate Error Estimate −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 Error Error −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 −8 −8 10 10 −8 −7 −6 −5 −4 −3 −2 −1 0 −8 −7 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Tolerance Tolerance Figure 1. True and estimated errors of the doubly quasi-consistent peer method applied to the test problems. A new approach to control the global error of numerical methods for differential equations – p.21/59

Global Error Estimation & Control DYNAMIC BEHAVIOUR OF THE ERRORS AND THE ESTIMATE: −5 First Test Problem x 10 −4 Second Test Problem x 10 6 3 Exact Error Exact Error Error Estimate Error Estimate Local Error Local Error 5 2.5 4 Accuracy in sup−norm 2 Accuracy in sup−norm 3 1.5 2 1 1 0.5 0 0 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 t t Figure 2. Numerical results obtained for the method when ǫ g = 10 − 04 . A new approach to control the global error of numerical methods for differential equations – p.22/59

Global Error Estimation & Control DISADVANTAGE of THE STEPSIZE SELECTION: ▽ A new approach to control the global error of numerical methods for differential equations – p.23/59

Global Error Estimation & Control DISADVANTAGE of THE STEPSIZE SELECTION: The fixed -stepsize methods are not efficient! ▽ A new approach to control the global error of numerical methods for differential equations – p.23/59

Global Error Estimation & Control DISADVANTAGE of THE STEPSIZE SELECTION: The fixed -stepsize methods are not efficient! Thus, the second issue is: Accommodation of Doubly Quasi-Consistent Numerical Schemes to variable meshes ▽ A new approach to control the global error of numerical methods for differential equations – p.23/59

Global Error Estimation & Control DISADVANTAGE of THE STEPSIZE SELECTION: The fixed -stepsize methods are not efficient! Thus, the second issue is: Accommodation of Doubly Quasi-Consistent Numerical Schemes to variable meshes This is to be done on the basis of: The Polynomial Interpolation Technique A new approach to control the global error of numerical methods for differential equations – p.23/59

EPP Methods of Interpolation Type We introduce a variable grid with a diameter τ on the integration interval [ t 0 , t end ] by w τ := { t k +1 = t k + τ k , k = 0 , 1 , . . . , K − 1 , t K = t end } where τ := max 0 ≤ k ≤ K − 1 { τ k } . It is clear that EPP -method (3) cannot be applied on w τ . ▽ A new approach to control the global error of numerical methods for differential equations – p.24/59

EPP Methods of Interpolation Type We introduce a variable grid with a diameter τ on the integration interval [ t 0 , t end ] by w τ := { t k +1 = t k + τ k , k = 0 , 1 , . . . , K − 1 , t K = t end } where τ := max 0 ≤ k ≤ K − 1 { τ k } . It is clear that EPP -method (3) cannot be applied on w τ . Let us consider that we have completed the ( k − 1) -th step of the size τ k − 1 and computed the numerical solution x k − 1 k − 1 ,i , i = 1 , 2 , . . . , s . ▽ A new approach to control the global error of numerical methods for differential equations – p.24/59

EPP Methods of Interpolation Type We introduce a variable grid with a diameter τ on the integration interval [ t 0 , t end ] by w τ := { t k +1 = t k + τ k , k = 0 , 1 , . . . , K − 1 , t K = t end } where τ := max 0 ≤ k ≤ K − 1 { τ k } . It is clear that EPP -method (3) cannot be applied on w τ . Let us consider that we have completed the ( k − 1) -th step of the size τ k − 1 and computed the numerical solution x k − 1 k − 1 ,i , i = 1 , 2 , . . . , s . Further, we want to advance the next step of the size τ k � = τ k − 1 . A new approach to control the global error of numerical methods for differential equations – p.24/59

EPP Methods of Interpolation Type At this point, we need two auxiliary grids: w k − 1 := { t k − 1 k − 1 ,i = t k + ( c i − 1) τ k − 1 , i = 1 , 2 , . . . , s } and w k := { t k k − 1 ,i = t k + ( c i − 1) τ k , i = 1 , 2 , . . . , s } where c i , i = 1 , 2 , . . . , s , are nodes of the fixed -stepsize EPP-method (3), which are considered to be distinct. ▽ A new approach to control the global error of numerical methods for differential equations – p.25/59

EPP Methods of Interpolation Type At this point, we need two auxiliary grids: w k − 1 := { t k − 1 k − 1 ,i = t k + ( c i − 1) τ k − 1 , i = 1 , 2 , . . . , s } and w k := { t k k − 1 ,i = t k + ( c i − 1) τ k , i = 1 , 2 , . . . , s } where c i , i = 1 , 2 , . . . , s , are nodes of the fixed -stepsize EPP-method (3), which are considered to be distinct. Now we utilize the interpolating polynomial H s − 1 k − 1 ( t ) of degree s − 1 fitted to the data x k − 1 k − 1 , i , i = 1 , 2 , . . . , s , from the most recent step to accommodate this numerical solution to the new stepsize τ k . A new approach to control the global error of numerical methods for differential equations – p.25/59

EPP Methods of Interpolation Type The scheme of computation is the following: ▽ A new approach to control the global error of numerical methods for differential equations – p.26/59

EPP Methods of Interpolation Type The scheme of computation is the following: 1. We calculate the new stage values x k k − 1 , i , i = 1 , 2 , . . . , s , for the grid w k by the polynomial H s − 1 k − 1 ( t ) . ▽ A new approach to control the global error of numerical methods for differential equations – p.26/59

EPP Methods of Interpolation Type The scheme of computation is the following: 1. We calculate the new stage values x k k − 1 , i , i = 1 , 2 , . . . , s , for the grid w k by the polynomial H s − 1 k − 1 ( t ) . 2. We compute the numerical solution x k ki , i = 1 , 2 , . . . , s , for the next step of the size τ k by formula (3). A new approach to control the global error of numerical methods for differential equations – p.26/59

EPP Methods of Interpolation Type DEFINITION 2: The EPP -method of the form t k x k k − 1 ,j = H s − 1 k − 1 ( t k k − 1 ,j = t k + ( c j − 1) τ k , k − 1 ,j ) , (5 a ) s s � � x k b ij x k a ij g ( t k k − 1 ,j , x k ki = k − 1 ,j + τ k k − 1 ,j ) , (5 b ) j =1 j =1 where H s − 1 k − 1 ( t ) is the interpolating polynomial of degree s − 1 fitted to the numerical solution x k − 1 k − 1 ,i , i = 1 , 2 , . . . , s , from the previous step is called the Explicit Parallel Peer method with polynomial interpolation of the numerical solution (or, briefly, the interpolating EPP-method ). A new approach to control the global error of numerical methods for differential equations – p.27/59

EPP Methods of Interpolation Type THEOREM 3: Let the EPP -method (3) with distinct nodes c i be zero-stable. Then the interpolating EPP-method (5) is zero-stable if and only if the following condition holds: � � m � � � BH ( θ k + m − l ) � ≤ R, for all k ≥ 0 and m ≥ 0 (6) � � � � � l =0 s ( c i − 1) θ k − c n +1 � where h ij ( θ k ) := , i, j = 1 , 2 , . . . , s, c j − c n n =1 , n � = j R is a finite constant and θ k := τ k /τ k − 1 is the corresponding stepsize ratio of the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.28/59

EPP Methods of Interpolation Type THEOREM 3: Let the EPP -method (3) with distinct nodes c i be zero-stable. Then the interpolating EPP-method (5) is zero-stable if and only if the following condition holds: � � m � � � BH ( θ k + m − l ) � ≤ R, for all k ≥ 0 and m ≥ 0 (6) � � � � � l =0 s ( c i − 1) θ k − c n +1 � where h ij ( θ k ) := , i, j = 1 , 2 , . . . , s, c j − c n n =1 , n � = j R is a finite constant and θ k := τ k /τ k − 1 is the corresponding stepsize ratio of the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.29/59

EPP Methods of Interpolation Type DEFINITION 3: The set of grids where the interpolating EPP -method (5) is stable is further referred to as the set ω 1 ,ω 2 ( t 0 , t end ) of admissible grids . Such grids satisfy the W ∞ condition 0 ≤ ω 1 < θ k < ω 2 ≤ ∞ , k = 0 , 1 , ..., K − 1 , (7) with constants ω 1 and ω 2 for which ω 1 ≤ 1 ≤ ω 2 . A new approach to control the global error of numerical methods for differential equations – p.30/59

EPP Methods of Interpolation Type DEFINITION 4: The fixed -stepsize EPP-method (3) is said to be strongly stable if its propagation matrix B has only one simple eigenvalue at one and all others lie in the open unit disc. ▽ A new approach to control the global error of numerical methods for differential equations – p.31/59

EPP Methods of Interpolation Type DEFINITION 4: The fixed -stepsize EPP-method (3) is said to be strongly stable if its propagation matrix B has only one simple eigenvalue at one and all others lie in the open unit disc. THEOREM 4: Let the underlying fixed-stepsize s -stage EPP-method (3) of consistency order p ≥ 0 and with distinct nodes c i be strongly stable. Then there exist constants ω 1 and ω 2 , satisfying (7) , such that the corresponding s -stage interpolating EPP-method (5) is stable on any grid from the set W ∞ ω 1 ,ω 2 ( t 0 , t end ) . A new approach to control the global error of numerical methods for differential equations – p.31/59

1 1 EPP Methods of Interpolation Type DEFINITION 5: The fixed -stepsize EPP-method (3) is said to be optimally stable if its propagation matrix B has only one simple eigenvalue at one and all others are zero. ▽ A new approach to control the global error of numerical methods for differential equations – p.32/59

EPP Methods of Interpolation Type DEFINITION 5: The fixed -stepsize EPP-method (3) is said to be optimally stable if its propagation matrix B has only one simple eigenvalue at one and all others are zero. THEOREM 5: Let the underlying fixed-stepsize s -stage 1 v T EPP-method (3) with distinct nodes c i be consistent of order p ≥ 0 . Suppose that its propagation matrix B 1 := (1 , 1 , . . . , 1) T and v := ( v 1 , v 2 , . . . , v s ) T . Then satisfies B = (8) where the corresponding s -stage interpolating EPP-method (5) is stable on any grid from the set W ∞ 0 , ∞ ( t 0 , t end ) . A new approach to control the global error of numerical methods for differential equations – p.32/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.33/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.34/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.35/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.36/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.37/59

EPP Methods of Interpolation Type THEOREM 6: Let the right -hand side of ODE (1) be max { p, s − 1 } times continuously differentiable in a neighborhood of the exact solution and the stable EPP-method (3) with distinct nodes c i be consistent of order p ≥ 1 . Suppose that the starting vector X 0 0 is known with an error of O ( τ min { p,s − 1 } ) and there exists a nonempty set W ∞ ω 1 ,ω 2 ( t 0 , t end ) of admissible grids with finite parameter ω 2 . Then the EPP-method (5) is convergent of order min { p, s − 1 } , i.e. its global error satisfies � X ( T k k ) − X k k � ≤ Cτ min { p,s − 1 } , k = 1 , 2 , . . . , K. where C is a finite constant. A new approach to control the global error of numerical methods for differential equations – p.38/59

EPP Methods of Interpolation Type REMARK 1: Additionally, Theorem 6 says that double quasi-consistency condition (2) does not work in general to improve the convergence order of interpolating EPP-methods (5) because of the variable matrix H ( θ k ) involved in numerical integration. ▽ A new approach to control the global error of numerical methods for differential equations – p.39/59

EPP Methods of Interpolation Type REMARK 1: Additionally, Theorem 6 says that double quasi-consistency condition (2) does not work in general to improve the convergence order of interpolating EPP-methods (5) because of the variable matrix H ( θ k ) involved in numerical integration. Further, we discuss how to accommodate double quasi-consistency to error estimation in interpolating EPP-methods . ▽ A new approach to control the global error of numerical methods for differential equations – p.39/59

EPP Methods of Interpolation Type REMARK 1: Additionally, Theorem 6 says that double quasi-consistency condition (2) does not work in general to improve the convergence order of interpolating EPP-methods (5) because of the variable matrix H ( θ k ) involved in numerical integration. Further, we discuss how to accommodate double quasi-consistency to error estimation in interpolating EPP-methods . We impose the following extra condition: τ/τ k ≤ Ω < ∞ , k = 0 , 1 , . . . , K − 1 , (9) where τ is the diameter of the grid. The set of grids satisfying (7) and (9) is denoted by W Ω ω 1 ,ω 2 ( t 0 , t end ) . A new approach to control the global error of numerical methods for differential equations – p.39/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.40/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.41/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.42/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.43/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.44/59

EPP Methods of Interpolation Type THEOREM 7: Let ODE (1) be sufficiently smooth and the stable EPP -method (3) of order p ≥ 1 and with distinct nodes c i be doubly quasi-consistent. Suppose that another solution ¯ X k k of order min { p + 1 , s } is known for a mesh w τ and the polynomial H s − 1 k − 1 ( t ) satisfies p ≤ s − 1 . (10) Then the interpolating EPP-method k = ( B ⊗ I m ) ¯ k − 1 , ¯ X k H s − 1 k − 1 ( T k k − 1 ) + τ k ( A ⊗ I m ) g ( T k H s − 1 k − 1 ( T k k − 1 )) where ¯ k − 1 ( t ) is fitted to the solution ¯ H s − 1 X k − 1 k − 1 , is doubly quasi-consistent on the grid w τ . A new approach to control the global error of numerical methods for differential equations – p.45/59

EPP Methods of Interpolation Type REMARK 2: If the more accurate numerical solution ¯ X k k in the formulation of Theorem 7 is computed by another s -stage interpolating EPP-method (5) then condition (10) must be replaced with the more stringent one p ≤ s − 2 (11) to retain the double quasi-consistency. ▽ A new approach to control the global error of numerical methods for differential equations – p.46/59

EPP Methods of Interpolation Type REMARK 2: If the more accurate numerical solution ¯ X k k in the formulation of Theorem 7 is computed by another s -stage interpolating EPP-method (5) then condition (10) must be replaced with the more stringent one p ≤ s − 2 (11) to retain the double quasi-consistency. Notice that utilization of another s -stage interpolating EPP-method (5) is a natural requirement of the embedded method error estimation presented by formula (4). ▽ A new approach to control the global error of numerical methods for differential equations – p.46/59

EPP Methods of Interpolation Type REMARK 2: If the more accurate numerical solution ¯ X k k in the formulation of Theorem 7 is computed by another s -stage interpolating EPP-method (5) then condition (10) must be replaced with the more stringent one p ≤ s − 2 (11) to retain the double quasi-consistency. Notice that utilization of another s -stage interpolating EPP-method (5) is a natural requirement of the embedded method error estimation presented by formula (4). Thus, Remark 2 allows the same numerical solution ¯ X k k to be used effectively in the doubly quasi-consistent method and in our error evaluation scheme as well. A new approach to control the global error of numerical methods for differential equations – p.46/59

Efficient Global Error Control CONSTRUCTION of EMBEDDED INTERPOLATING EPP -METHODS: It follows from Theorem 7 and Remark 2 that the embedded s -stage underlying fixed-stepsize EPP-methods (3) must be of consistency orders s − 3 and s . ▽ A new approach to control the global error of numerical methods for differential equations – p.47/59

Efficient Global Error Control CONSTRUCTION of EMBEDDED INTERPOLATING EPP -METHODS: It follows from Theorem 7 and Remark 2 that the embedded s -stage underlying fixed-stepsize EPP-methods (3) must be of consistency orders s − 3 and s . the lower order method is to be doubly quasi-consistent of order s − 2 and, hence, it is convergent of the same order on equidistant meshes. A new approach to control the global error of numerical methods for differential equations – p.47/59

Efficient Global Error Control CONSTRUCTION of EMBEDDED INTERPOLATING EPP -METHODS (cont.): We fit the interpolating polynomial to the numerical solution obtained from the higher order embedded formula and denote it further by ¯ H s − 1 k − 1 ( t ) . ▽ A new approach to control the global error of numerical methods for differential equations – p.48/59

Efficient Global Error Control CONSTRUCTION of EMBEDDED INTERPOLATING EPP -METHODS (cont.): We fit the interpolating polynomial to the numerical solution obtained from the higher order embedded formula and denote it further by ¯ H s − 1 k − 1 ( t ) . Our error estimation formula is presented by � ¯ ∆ 1 X k X k � k = ( B emb − B ) ⊗ I m k − 1 + k − 1 , ¯ � � g ( T k X k + τ k ( A emb − A ) ⊗ I m k − 1 ) where A , B and A emb , B emb are coefficients of the EPP-methods of orders s − 2 and s − 1 , respectively. A new approach to control the global error of numerical methods for differential equations – p.48/59

1 Efficient Global Error Control In this way, we derive three pairs of embedded interpolating EPP -methods of orders s − 2 and s − 1 abbreviated further as IEPP23 , IEPP34 and IEPP45 . ▽ A new approach to control the global error of numerical methods for differential equations – p.49/59

Efficient Global Error Control In this way, we derive three pairs of embedded interpolating EPP -methods of orders s − 2 and s − 1 abbreviated further as IEPP23 , IEPP34 and IEPP45 . 1 v T All these numerical schemes satisfy the following conditions imposed on their coefficients: B emb = B = and c emb = c. Thus, IEPP23 , IEPP34 and IEPP45 are determine completely by fixing two matrices A , A emb and two vectors c and v . A new approach to control the global error of numerical methods for differential equations – p.49/59

Efficient Global Error Control In this way, we derive three pairs of embedded interpolating EPP -methods of orders s − 2 and s − 1 abbreviated further as IEPP23 , IEPP34 and IEPP45 . 1 v T All these numerical schemes satisfy the following conditions imposed on their coefficients: B emb = B = and c emb = c. Thus, IEPP23 , IEPP34 and IEPP45 are determine completely by fixing two matrices A , A emb and two vectors c and v . A new approach to control the global error of numerical methods for differential equations – p.50/59

Efficient Global Error Control NUMERICAL RESULTS for our Test Problems: Accuracy Graph for Test Problem I Accuracy Graph for Test Problem II −2 −2 10 10 IEPP23 IEPP23 IEPP34 IEPP34 −3 IEPP45 10 IEPP45 −3 10 −4 10 −4 10 −5 10 Global Error Error −5 −6 10 10 −7 10 −6 10 −8 10 −7 10 −9 10 −8 −10 10 10 −8 −7 −6 −5 −4 −3 −2 −8 −7 −6 −5 −4 −3 −2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Tolerance Tolerance Figure 3. Exact errors of the embedded peer schemes with built -in our error estimation. A new approach to control the global error of numerical methods for differential equations – p.51/59

Efficient Global Error Control NUMERICAL RESULTS for our Test Problems: Accuracy Graph for Test Problem I Accuracy Graph for Test Problem II 0 −1 10 10 ODE23 ODE23 ODE45 ODE45 −1 10 −2 ODE113 ODE113 10 −2 10 −3 10 −3 10 Global Error −4 10 Error −4 10 −5 10 −5 10 −6 10 −6 10 −7 10 −7 10 −8 −8 10 10 −8 −7 −6 −5 −4 −3 −2 −8 −7 −6 −5 −4 −3 −2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Tolerance Tolerance Figure 4. Exact errors of all explicit MatLab solvers with relative error control set by "RelTol"="AbsTol". A new approach to control the global error of numerical methods for differential equations – p.52/59

Efficient Global Error Control NUMERICAL RESULTS for our Test Problems: Accuracy Graph for Test Problem I Accuracy Graph for Test Problem II 0 −1 10 10 ODE23 ODE23 ODE45 ODE45 −1 10 −2 ODE113 ODE113 10 −2 10 −3 10 −3 10 Global Error −4 10 Error −4 10 −5 10 −5 10 −6 10 −6 10 −7 10 −7 10 −8 −8 10 10 −8 −7 −6 −5 −4 −3 −2 −8 −7 −6 −5 −4 −3 −2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Tolerance Tolerance Figure 5. Exact errors of all explicit MatLab solvers without relative error control set by "RelTol" := 1 . 0 E − 10 . A new approach to control the global error of numerical methods for differential equations – p.53/59