Continuous hyperbolic boundary value problems : a brief introduction Discretized problems : zero initial data Stability of finite difference schemes for hyperbolic initial boundary value problems II 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
Continuous hyperbolic boundary value problems : a brief introduction Discretized problems : zero initial data Plan of the second course Continuous hyperbolic boundary value problems : a brief introduction 1 The one-dimensional case The multi-dimensional case Discretized problems : zero initial data 2 The discretized boundary value problem : strong stability The normal modes analysis The Uniform Kreiss-Lopatinskii Condition The main stability result Summary J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Plan Continuous hyperbolic boundary value problems : a brief introduction 1 The one-dimensional case The multi-dimensional case Discretized problems : zero initial data 2 The discretized boundary value problem : strong stability The normal modes analysis The Uniform Kreiss-Lopatinskii Condition The main stability result Summary J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Consider the initial boundary value problem : ∂ t u + A ∂ x u = F ( t , x ) , in [0 , T ] × R + , B u ( t , 0) = g ( t ) , in [0 , T ] , u | t =0 = f , on R + . Space domain : R + . Boundary : { x = 0 } . Linear system with constant coefficients : A ∈ M N ( R ), u ∈ R N , B ∈ M p , N ( R ). General problem Which boundary conditions B give a well-posed problem ? The solution and its trace should be estimated in the same functional spaces as the data. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics We assume that the operator ∂ t + A ∂ x is hyperbolic : A is diagonalizable. We introduce a set of eigenvalues and eigenvectors : 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 N � F ( t , x ) = F j ( t , x ) r j . j =1 J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics The problem reads ∂ t α j + λ j ∂ x α j = F j ( t , x ) , in [0 , T ] × R + , � j α j ( t , 0) B r j = g ( t ) , in [0 , T ] , α j | t =0 = β j , on R + . This system can be solved by the method of characteristics, separating outgoing characteristics from incoming characteristics. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics Definition The eigenvalue λ j corresponds to an outgoing characteristic if λ j < 0, and to an incoming characteristic if λ j > 0. The analysis is in two steps. For simplicity, we shall assume that A is invertible. This is the so-called noncharacteristic case. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics We start with outgoing characteristics : λ j < 0. Then we find � t α j ( t , x ) = β j ( x − λ j t ) + F j ( s , x − λ j ( t − s )) d s , 0 and this formula defines α j in R + × R + since λ j < 0. In particular, the trace α j ( t , 0) can be explicitly computed from the data : � t α j ( t , 0) = β j ( − λ j t ) + F j ( s , − λ j ( t − s )) d s . 0 J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics We go on with incoming characteristics : λ j > 0. We find � t α j ( t , x ) = β j ( x − λ j t ) + F j ( s , x − λ j ( t − s )) d s , 0 in the domain { ( t , x ) ∈ R + × R + / x ≥ λ j t } , and � t α j ( t , x ) = α j ( t − x /λ j , 0) + F j ( s , x − λ j ( t − s )) d s , t − x /λ j in the domain { ( t , x ) ∈ R + × R + / x < λ j t } . The trace α j ( t , 0) is not determined by the data ! J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics To determine the solution completely, the boundary condition N � α j ( t , 0) B r j = g ( t ) , j =1 should determine the trace of incoming characteristics in terms of the source term g and of the trace of outgoing characteristics. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case The method of characteristics To determine the solution completely, the boundary condition N � α j ( t , 0) B r j = g ( t ) , j =1 should determine the trace of incoming characteristics in terms of the source term g and of the trace of outgoing characteristics. Conclusion The problem can be well-posed only if � � R p = Span B r 1 , . . . , B r q , with the convention λ 1 , . . . , λ q > 0, λ q +1 , . . . , λ N < 0. This implies p ≤ q . J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Conclusion In the one-dimensional case, the initial boundary value problem is well-posed (existence/uniqueness) if and only if The number p of independent boundary conditions is equal to the 1 number of incoming characteristics ( q = p ), There holds 2 � � Ker B ∩ Span r 1 , . . . , r p = { 0 } , where r 1 , . . . , r p span the eigenspace of A associated with positive eigenvalues. Observe that the latter relation is compatible with dimensions ! In particular, there always exists a matrix B for which the problem is well-posed. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Plan Continuous hyperbolic boundary value problems : a brief introduction 1 The one-dimensional case The multi-dimensional case Discretized problems : zero initial data 2 The discretized boundary value problem : strong stability The normal modes analysis The Uniform Kreiss-Lopatinskii Condition The main stability result Summary J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Consider the initial boundary value problem : L ( ∂ ) u := ∂ t u + � d in [0 , T ] × R d j =1 A j ∂ x j u = F , + , on [0 , T ] × R d − 1 , B u | x d =0 = g , on R d u | t =0 = f , + . Space domain : half-space R d + = { x d > 0 } . Linear system with constant coefficients . General problem The differential operator L ( ∂ ) is given. Can we find/characterize the boundary conditions B that give a well-posed problem ? J.-F. Coulombel Fully discrete hyperbolic boundary value problems
Continuous hyperbolic boundary value problems : a brief introduction The one-dimensional case Discretized problems : zero initial data The multi-dimensional case Key points of the analysis In the non-characteristic case ( A d invertible), the analysis relies on three major assumptions : A stability assumption for the Cauchy problem : hyperbolicity. 1 An additional structural assumption on the symbol associated with 2 the Cauchy problem (geometric regularity of eigenelements). A compatibility condition between the boundary conditions and the 3 hyperbolic system : the Uniform Kreiss-Lopatinskii Condition. J.-F. Coulombel Fully discrete hyperbolic boundary value problems
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