Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary High Order 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. Advances in Applied Mathematics in memoriam of Professor Saul Abarbanel Tel Aviv University, December 18 - 20, 2018. High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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. • Summary. High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 . High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary Semi-discrete approximations for PDEs. � ∂ � Let Q be the discretization of P where we assume: ∂ x Assumption 1 : Q is semibound in some equivalent scalar product ( · , · ) H = ( · , H · ) , i.e. ( w , Q w ) H ≤ α ( w , w ) H = α � w � 2 H High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary Semi-discrete approximations for PDEs. � ∂ � Let Q be the discretization of P where we assume: ∂ x Assumption 1 : 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 2 : 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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. High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 . High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary Fully-discrete approximations for PDEs or ODEs. We assume: Assumption 1 : In some equivalent norm � · � H � Q � H ≤ 1 + α ∆ t High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary 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 (Strang and Fix) � � � E n � H = O � T µ � H . High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
Outline Introduction Ordinary Differential Equations. Examples The Heat Equations Post–processing Summary Fully-discrete approximations for PDEs or ODEs. Observation ∂ E = Q E + ∆ t T e and E n + 1 = QE n + T n ∂ t are exact while � E � H ( t ) ≤ � E � H ( 0 ) e α t + e α t − 1 N →∞ 0 ≤ τ ≤ t � T e � H max − − − − → 0 α and � E n � H ≤ � E 0 � H e α t + e α t − 1 N →∞ 0 ≤ µ ≤ 0 � T µ � H max − − − − → 0 α are estimates! High Order Error Inhibiting Schemes for Differential Equations Adi Ditkowski
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