Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Stability of finite difference schemes for hyperbolic initial boundary value problems I Jean-Fran¸ cois Coulombel Laboratoire Paul Painlev´ e (UMR CNRS 8524) CNRS, Universit´ e Lille 1 Team Project SIMPAF - INRIA Lille Nord Europe Nonlinear hyperbolic PDEs, dispersive and transport equations, Trieste, June 2011 J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Plan of the first course Hyperbolic equations in one space dimension : a brief introduction 1 Discretized equations : stability and convergence 2 Some facts on the fully discrete Cauchy problem Some facts on power bounded matrices What about convergence ? Summary J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence We consider the one-dimensional Cauchy problem for a first order system : � in [0 , T ] × R , ∂ t u + A ∂ x u = 0 , on R . u | t =0 = f , Space domain R , A ∈ M N ( R ), u ( t , x ) ∈ R N . Linear system with constant coefficients (for simplicity). Question Under which condition is the Cauchy problem well-posed ? (Existence, uniqueness and continuous dependence of the solution on the initial condition. Of course, this heavily depends on the functional framework, as usual in the study of partial differential equations.) J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Hyperbolicity Answer by Fourier transform : � ∂ t � u + i ξ A � u = 0 , in [0 , T ] , u (0 , ξ ) = � on R . f ( ξ ) , � This gives the formula u ( t , ξ ) = exp( − i t ξ A ) � � f ( ξ ) . J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Hyperbolicity Definition (hyperbolicity) The operator ∂ t + A ∂ x is said to be hyperbolic if sup � exp( i η A ) � < + ∞ . η ∈ R J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Hyperbolicity Definition (hyperbolicity) The operator ∂ t + A ∂ x is said to be hyperbolic if sup � exp( i η A ) � < + ∞ . η ∈ R Proposition (easy case of a more general result by Kreiss) The operator ∂ t + A ∂ x is hyperbolic if and only if the matrix A is diagonalizable with real eigenvalues. In this case, the Cauchy problem is well-posed in L 2 ( R ) : for all f ∈ L 2 ( R ), there exists a unique solution u ∈ C ( R t ; L 2 ( R x )), and this solution satisfies the estimate sup � u ( t , · ) � L 2 ( R ) ≤ C 0 � f � L 2 ( R ) , t ∈ R for a certain numerical constant C 0 > 0. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Integration along characteristics We consider the eigenvalues and eigenvectors of A : λ j , r j , j = 1 , . . . , N . The solution u is decomposed on the basis ( r 1 , . . . , r N ) : N � u ( t , x ) = α j ( t , x ) r j , j =1 N � f ( x ) = β j ( x ) r j . j =1 The system of PDEs decouples into � in [0 , T ] × R , ∂ t α j + λ j ∂ x α j = 0 , on R . α j | t =0 = β j , J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence Integration along characteristics Each function α j solves a scalar transport equation, which can be solved by the method of characteristics : α j ( t , x ) = β j ( x − λ j t ) . This gives the explicit formula � N u ( t , x ) = β j ( x − λ j t ) r j . j =1 Corollary If A is diagonalizable with real eigenvalues, then the Cauchy problem is also well-posed in any L p ( R ), 1 ≤ p < + ∞ . This property is specific to one-dimensional problems . J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Hyperbolic equations in one space dimension : a brief introduction Discretized equations : stability and convergence To remember • We only consider the case of hyperbolic systems ( A diagonalizable) : there is an explicit formula for the solution (in particular, finite speed of propagation.) • When the initial condition belongs to L 2 ( R ), there holds u ( t , ξ ) = exp( − i t ξ A ) � � f ( ξ ) . • The L 2 well-posedness theory is the only one that extends to general systems in several space dimensions (Brenner, Rauch...). This is the reason why we do not consider here the well-posedness theory in BV ( R ). The ultimate goal is to get results in any space dimension. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Plan Hyperbolic equations in one space dimension : a brief introduction 1 Discretized equations : stability and convergence 2 Some facts on the fully discrete Cauchy problem Some facts on power bounded matrices What about convergence ? Summary J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Discretizing the Cauchy problem We still consider the Cauchy problem � in [0 , + ∞ [ × R , ∂ t u + A ∂ x u = 0 , u | t =0 = f . Our goal is to construct an approximation the solution u ( t , x ). ∆ t , ∆ x : time and space steps. The ratio λ = ∆ t / ∆ x is kept fixed , and ∆ t is allowed to be small (∆ x varies accordingly). λ is called the Courant-Friedrichs-Lewy number (CFL). J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Discretizing the Cauchy problem In what follows, we let U n j denote the approximation of the solution u on the cell [ n ∆ t , ( n + 1) ∆ t [ × [ j ∆ x , ( j + 1) ∆ x [, with n ∈ N and j ∈ Z . U n j is not necessarily a pointwise approximation of u ( t n , x j ). If U ∆ denotes the corresponding step function, the approximation should be understood in the following sense : � u − U ∆ � L ∞ ([0 , T ]; L 2 ( R )) = o (1) . (Observe that U ∆ does not belong to C ( L 2 ) but only to L ∞ ( L 2 ). Continuity is only recovered in the limit ∆ t → 0.) J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Numerical scheme for the Cauchy problem A numerical scheme with one time step reads : � U n +1 = Q U n j ∈ Z , j , j U 0 j ∈ Z , j = f j , with a discretized evolution operator p � A ℓ T ℓ , Q := ( TU ) j := U j +1 . ℓ = − r The scheme involves “ r points on the left, and p points on the right”. Usually, the matrices A ℓ are polynomial functions of λ A . J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Numerical scheme for the Cauchy problem • Here we only consider linear schemes : the mapping U 0 �→ U 1 is linear. The matrices A ℓ do not depend on j , n nor on the initial condition U 0 . • More elaborate schemes (flux limiters, ENO, WENO...) are non-linear ! Their analysis may be much more complicated. • One possible discretization of the initial condition is � ( j +1) ∆ x 1 f j := f ( y ) d y . ∆ x j ∆ x Good stability property (use Cauchy-Schwarz) : � ∆ x | f j | 2 ≤ � f � L 2 ( R ) . j ∈ Z J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Some facts on the fully discrete Cauchy problem Hyperbolic equations in one space dimension : a brief introduction Some facts on power bounded matrices Discretized equations : stability and convergence What about convergence ? Summary Fundamental examples • The upwind scheme : for a scalar transport equation j − λ U n +1 = U n 2 ( a + | a | ) ( U n j − U n j − 1 ) j − λ 2 ( a − | a | ) ( U n j +1 − U n j ∈ Z , j ) , U 0 j ∈ Z . j = f j , J.-F. Coulombel Fully discrete hyperbolic boundary value problems
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