Error Inhibiting Schemes for Differential Equations Adi Ditkowski - PowerPoint PPT Presentation

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Error Inhibiting Schemes for Differential Equations Adi Ditkowski Department of Applied Mathematics Tel Aviv University Joint work with Sigal Gottlieb, Chi-Wang Shu and Paz Fink. ICERM, August 2018. Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Outline of the talk: • Review of the classical theory. • Semi-discrete approximations for PDEs. • Fully-discrete approximations for PDEs or ODEs. • Error Inhibiting Schemes for ODEs. • Error Inhibiting Schemes for PDEs. • Block Finite Difference schemes for the Heat equation. Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. Review of the classical theory Semi-discrete approximations for PDEs. Consider the differential problem: � ∂ � ∂ u x ∈ Ω ⊂ R d , t ≥ 0 = P u , ∂ t ∂ x u ( t = 0 ) = f . It is assumed that this problem is well posed, In particular ∃ K ( t ) < ∞ s.t. || u ( t ) || ≤ K ( t ) || f || . Typically K ( t ) = Ke α t . Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. � ∂ � Let Q be the discretization of P where we assume: ∂ x Assumption 1 : The discrete operator Q is based on the the grid points { x j } , j = 1 , . . . , N . Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. � ∂ � Let Q be the discretization of P where we assume: ∂ x Assumption 1 : The discrete operator Q is based on the the grid points { x j } , j = 1 , . . . , N . Assumption 2 : Q is semibound in some equivalent scalar product ( · , · ) H = ( · , H · ) , i.e. ( w , Q w ) H ≤ α ( w , w ) H = α � w � 2 H Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. � ∂ � Let Q be the discretization of P where we assume: ∂ x Assumption 1 : The discrete operator Q is based on the the grid points { x j } , j = 1 , . . . , N . Assumption 2 : Q is semibound in some equivalent scalar product ( · , · ) H = ( · , H · ) , i.e. ( w , Q w ) H ≤ α ( w , w ) H = α � w � 2 H Assumption 3 : The local truncation error of Q is T e and is defined by T e = P w − Q w , where w ( x ) is a smooth function and w is the N →∞ projection of w ( x ) onto the grid. T e − − − − → 0 Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. Example: ∂ 2 u ∂ u = ∂ x 2 + F ( x , t ) , x ∈ [ 0 , 2 π ) , t ≥ 0 ∂ t u ( t = 0 ) = f ( x ) with periodic boundary conditions. Consider the approximation:  ... ... ...  1   1 1 − 2 1   u xx ≈ u   h 2 1 − 2 1     ... ... ... 1 = D + D − u . Then ( T e ) j = h 2 � � xxxx + O ( h 4 ) ( w , D + D − w ) ≤ 0 u j and 12 Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. Consider the semi–discrete approximation: ∂ v = Q v , t ≥ 0 ∂ t v ( t = 0 ) = f . Proposition : Under Assumptions 1–3 The semi–discrete approximation converges. Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. Proposition : Under Assumptions 1–3 The semi–discrete approximation converges. Proof : Let u is the projection of u ( x , t ) onto the grid. Then ∂ u = P u = Q u + T e ∂ t ∂ v = Q v ∂ t Let E = u − v then ∂ E = Q E + T e ∂ t Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Semi-discrete approximations for PDEs. ∂ E = Q E + T e ∂ t By taking the H scalar product with E : � E , ∂ E � 1 ∂ ∂ = ∂ t ( E , E ) H = � E � H ∂ t || E � H ∂ t 2 H = ( E , Q E ) H + ( E , T e ) H α � E � 2 ≤ H + � E � H � T e � H Thus ∂ ∂ t � E � H ≤ α � E � H + � T e � H Therefore: � E � H ( t ) ≤ � E � H ( 0 ) e α t + e α t − 1 N →∞ 0 ≤ τ ≤ t � T e � H max − − − − → 0 α Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. Fully-discrete approximations for PDEs or ODEs. Consider the differential problem: ∂ u = P u ∂ t u ( t = 0 ) = f . It is assumed that this problem is well posed, In particular ∃ K ( t ) < ∞ s.t. || u ( t ) || ≤ K ( t ) || f || . Typically K ( t ) = Ke α t . Remark : in order to simplify the explanation we consider the constant coefficients P . Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. Consider the multistep approximation: p � v n + 1 = Q j v n − j j = 0 where t n = n ∆ t and v n is the approximation to u ( t n ) . Denoting: ( u ( t n ) , u ( t n − 1 ) , ..., u ( t n − p )) T = U n ( v n , v n − 1 , ..., v n − p ) T . = V n Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. The scheme can be written as  Q 0 Q 1 ... Q n − p  I     0 I V n + 1 = V n = Q V n     ...     0 ... I 0 Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. We assume: Assumption 1 : In some equivalent norm � · � H � Q � H ≤ 1 + α ∆ t Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. We assume: Assumption 1 : In some equivalent norm � · � H � Q � H ≤ 1 + α ∆ t Assumption 2 : The local truncation error of Q is T n which is defined by ∆ tT n = W n + 1 − QW n where W n + 1 is the solution of the PDE/ODE whoe ’initial condition’ is W n at t n . It is assumed that N →∞ T n − − − − → 0 Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. Similar to the semi-discrete case U n + 1 = QU n + ∆ tT n V n + 1 = QV n Let E n = U n − V n then E n + 1 = QE n + ∆ tT n Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. Denoting by (= Q n − ν V ν for constant coefficients ) V n = S ∆ t ( t n , t ν ) V ν Then, using the discrete Duhamel’s principle n − 1 � E n = S ∆ t ( t n , 0 ) E 0 + ∆ t S ∆ t ( t n , t ν + 1 ) T ν , ν = 0 or, equivalently n − 1 E n = Q n E 0 + ∆ t � Q n − ν − 1 T ν . ν = 0 Therefore, using � Q µ � H ≤ ( 1 + α ∆ t ) µ ≈ e α t µ : � E n � H ≤ � E 0 � H e α t + e α t − 1 N →∞ 0 ≤ µ ≤ 0 � T µ � H max − − − − → 0 α Error Inhibiting Schemes for Differential Equations Adi Ditkowski

Outline Introduction Preliminary example Ordinary Differential Equations. Examples The Heat Equations Post–processing Fully-discrete approximations for PDEs or ODEs. Indeed, for all the classical schemes, e.g. ODE PDE Euler Forward Euler Backward Euler Backward Euler Trapezoid Lax–Friedrichs Multistep methods Lax–Wendroff Runge–Kutta methods Crank–Nicholson Leap–Frog Compact schemes Deferred–correction methods FE (see Strang and Fix) � � � E n � H = O � T µ � H . Error Inhibiting Schemes for Differential Equations Adi Ditkowski