Numerical Solutions to Partial Differential Equations Zhiping Li - PowerPoint PPT Presentation

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Evolution Equations and Their FDM Initial-Boundary Value Problems of Evolution Equations ∂ u ∂ t ( x , t ) = L ( u ( x , t )) + f ( x , t ) , ∀ ( x , t ) ∈ Ω × (0 , t max ] , g ( u ( x , t )) = g 0 ( x , t ) , ∀ ( x , t ) ∈ ∂ Ω 1 × (0 , t max ] , u ( x , 0) = u 0 ( x ) , ∀ x ∈ Ω , where L ( · ) is a (linear) differential operator acting on u with respect to x , and is assumed not explicitly depend on the time t . 2 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Evolution Equations and Their FDM Initial-Boundary Value Problems of Evolution Equations Definition An initial value problem is said to be well posed with respect to the norm � · � of a Banach space X , if it holds 1 for any given initial data u 0 ∈ X , i.e. � u 0 � < ∞ , there exists a solution; 2 there is a constant C > 0, such that, if v , w are the solutions of the problem with initial data v 0 , w 0 ∈ X respectively, then � v ( · , t ) − w ( · , t ) � ≤ C � v 0 ( · ) − w 0 ( · ) � , ∀ t ∈ [0 , t max ] . Remark: Similarly, we can define the well-posedness for the initial-boundary value problems. 3 / 30

Uniformly Well Conditioned Difference Schemes We consider the difference scheme of the following form B 1 U m +1 = B 0 U m + F m . 1 The difference operators B 1 and B 0 are independent of m . ∀ j ′ ∈ J Ω , α = 0 , 1. j ∈ J Ω b α 2 ( B α U m ) j ′ = � j ′ , j U m j , 3 F m contains information on the inhomogeneous boundary conditions as well as the source term of the difftl. eqn.. 4 B 1 = O ( τ − 1 ), B 1 is invertible, and B − 1 is uniformly well 1 conditioned, i.e. there exists a constant K > 0, such that � B − 1 1 � ≤ K τ. 5 Under the above assumptions, we can rewrite the difference scheme as U m +1 = B − 1 [ B 0 U m + F m ] . 1 Remark: In vector case of dimension p , U m ∈ R p , and b α j ′ , j ∈ R p × p . j

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Consistency, Order of Accuracy and Convergence Truncation Error of Difference Schemes Definition Suppose that u is an exact solution to the problem, define T m := B 1 u m +1 − [ B 0 u m + F m ] , as the truncation error of the scheme for the problem. Remark: By properly scaling the coefficients of the scheme (so that F m = f m ), the definition is consistent with T m := { [ B 1 (∆ + t + 1) − B 0 ] − [ ∂ t − L ] } u m . 5 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Consistency, Order of Accuracy and Convergence Consistency of Difference Schemes Definition The difference scheme is said to be consistent with the problem, if for all sufficiently smooth exact solution u of the problem, the truncation error satisfies T m → 0 , as τ ( h ) → 0 , ∀ m τ ≤ t max , ∀ j ∈ J Ω . j In particular, the difference scheme is said to be consistent with the problem in the norm � · � , if m − 1 � � T l � → 0 , τ as τ ( h ) → 0 , ∀ m τ ≤ t max . l =0 6 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Consistency, Order of Accuracy and Convergence Order of Accuracy of Difference Schemes Definition The difference scheme is said to have order of accuracy p in τ and q in h , if p and q are the largest integers so that j | ≤ C [ τ p + h q ] , | T m as τ ( h ) → 0 , ∀ m τ ≤ t max , ∀ j ∈ J Ω , for all sufficiently smooth solutions to the problem, where C is a constant independent of j , τ and h . Remark: Similarly, we can define the order of accuracy of a scheme with respect to a norm � · � , if m − 1 � T l � ≤ C [ τ p + h q ] , � τ as τ ( h ) → 0 , ∀ m τ ≤ t max . l =0 7 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Consistency, Order of Accuracy and Convergence Convergence of Difference Schemes Definition The difference scheme is said to be convergent in the norm � · � with respect to the problem, if the difference solution U m to the scheme satisfies � U m − u m � → 0 , as τ ( h ) → 0 , m τ → t ∈ [0 , t max ] , for all initial data u 0 with which the problem is well posed in the norm � · � . Remark: Similarly, we can define the order of convergence of a scheme to be p in τ and q in h with respect to the norm � · � , if � U m − u m � ≤ C [ τ p + h q ] , as τ ( h ) → 0 , m τ → t ∈ [0 , t max ] . 8 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Stability and Lax Equivalence Theorem Lax-Richtmyer Stability of Finite Difference Schemes Definition A finite difference scheme is said to be stable with respect to a norm � · � and a given refinement path τ ( h ), if there exists a constant K 1 > 0 independent of τ , h , V 0 and W 0 , such that � V m − W m � ≤ K 1 � V 0 − W 0 � , ∀ m τ ≤ t max , as long as V m − W m is a solution to the homogeneous difference scheme (meaning F m = 0 for all m ) with initial data V 0 − W 0 . 9 / 30

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Stability and Lax Equivalence Theorem Stability and Uniform Well-Posedness of Finite Difference Schemes The stability of finite difference schemes are closely related to the uniform well-posedness of the corresponding discrete problems. For linear problems, since V m − W m = B − 1 1 B 0 ( V m − 1 − W m − 1 ), thus, a scheme is (Lax-Richtmyer) stable if and only if � m � ≤ K 1 , B − 1 � � 1 B 0 ∀ m τ ≤ t max . Remark: In general, the uniform well-posedness of the scheme with respect to the boundary data and source term can be derived from the Lax-Richtmyer stability and the uniform invertibility of the scheme. 10 / 30

Lax Equivalence Theorem We always assume below that, for any initial data u 0 with which the corresponding problem is well posed, there exists an sequence of sufficiently smooth solutions v α , s.t. lim α →∞ � v 0 α − u 0 � → 0. Theorem For a uniformly solvable linear finite difference scheme which is consistent with a well-posed linear evolution problem, the stability is a necessary and sufficient condition for its convergence. 1 The original continuous problem must be well-posed. 2 Linear and linear, can’t emphasize more! Not true if nonlinear. 3 The consistency is also a crucial condition. 4 Do not forget uniform solvability of the scheme. 5 ”Stability ⇔ Convergence” (not hold if miss any condition).

Proof of Sufficiency of the Lax Equivalence Theorem (stability ⇒ convergence) If u ∈ X is a sufficiently smooth solution of the problem, then, by the definition of the truncation error, we have B 1 ( U m +1 − u m +1 ) = B 0 ( U m − u m ) − T m , or equivalently U m +1 − u m +1 = ( B − 1 1 B 0 )( U m − u m ) − B − 1 1 T m . Recursively, and by assuming U 0 = u 0 , we obtain m − 1 U m − u m = − � ( B − 1 1 B 0 ) l B − 1 1 T m − l − 1 . l =0 Thus, by the uniform solvability � B − 1 1 � ≤ K τ and the stability 1 B 0 ) l � ≤ K 1 , we have � U m − u m � ≤ KK 1 τ � m − 1 � ( B − 1 l =0 � T l � , ∀ m > 0. Therefore, by the definition of the consistency, we have τ ( h ) → 0 � U m − u m � = 0 , 0 ≤ m τ ≤ t max . lim

Proof of Sufficiency of the Lax Equivalence Theorem (continue) For a general solution u , let v α be the smooth solution sequence satisfying lim α →∞ � v 0 α − u 0 � → 0. 1 ∀ ε > 0, ∃ A > 0, such that � v 0 α − u 0 � < ε for all α > A . 2 For fixed β > A , let V m β be the solution of the difference scheme with V 0 β = v 0 β . 3 ∀ ε > 0, ∃ h ( ε ) > 0, s.t. � V m β − v m β � < ε , for all h < h ( ε ). Thus, by the stability and the uniform invertibility of the scheme and the well-posedness of the problem that, if h < h ( ε ), then � U m − u m � ≤ � U m − V m β � + � V m β − v m β � + � v m β − u m � ≤ ( K 1 + 1 + C ) ε. Since ε is arbitrary, this implies τ ( h ) → 0 � U m − u m � = 0 , 0 ≤ m τ ≤ t max . lim

More on Consistency, Stability and Convergence Consistency, Stability and Convergence of FDM Stability and Lax Equivalence Theorem Proof of Necessity of the Lax Equivalence Theorem (convergence ⇒ stability) 1 ( B − 1 1 B 0 ) m h : bounded linear operators in ( X , � · � ). 2 ( B − 1 h ≻ ( B − 1 h , either h < ˆ h , or h = ˆ 1 B 0 ) m 1 B 0 ) ˆ m h and m > ˆ m . ˆ 3 By the resonance theorem, if, for each given u 0 ∈ X , S t max ( u 0 ) := � ( B − 1 h u 0 � � 1 B 0 ) m � sup < ∞ , { h > 0 , m ≤ τ − 1 t max } then the sequence { ( B − 1 1 B 0 ) m h } is uniformly bounded, and consequently the scheme is stable. Suppose for some u 0 ∈ X , ∃ τ k → 0, m k τ k → t ∈ [0 , t max ], s.t. k →∞ � ( B − 1 1 B 0 ) m k h k u 0 � = ∞ . lim 14 / 30