Numerical Solutions to Partial Differential ... Numerical Solutions to Partial Di erential Equations

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Transcript of Numerical Solutions to Partial Differential ... Numerical Solutions to Partial Di erential Equations

  • 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, tmax],

    g(u(x , t)) = g0(x , t), ∀ (x , t) ∈ ∂Ω1 × (0, tmax], u(x , 0) = u0(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.

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  • 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 u0 ∈ X, i.e. ‖u0‖ 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, tmax].

    Remark: Similarly, we can define the well-posedness for the initial-boundary value problems.

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  • Uniformly Well Conditioned Difference Schemes

    We consider the difference scheme of the following form

    B1U m+1 = B0U

    m + Fm.

    1 The difference operators B1 and B0 are independent of m.

    2 (BαU m)j′ =

    ∑ j∈JΩ b

    α j′,jU

    m j , ∀ j′ ∈ JΩ, α = 0, 1.

    3 Fm contains information on the inhomogeneous boundary conditions as well as the source term of the difftl. eqn..

    4 B1 = O(τ −1), B1 is invertible, and B

    −1 1 is uniformly well

    conditioned, i.e. there exists a constant K > 0, such that

    ‖B−11 ‖ ≤ Kτ. 5 Under the above assumptions, we can rewrite the difference

    scheme as Um+1 = B−11 [B0U m + Fm] .

    Remark: In vector case of dimension p, Umj ∈ Rp, and bαj′,j ∈ Rp×p.

  • 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

    Tm := B1u m+1 − [B0um + Fm] ,

    as the truncation error of the scheme for the problem.

    Remark: By properly scaling the coefficients of the scheme (so that Fm = f m),

    the definition is consistent with Tm := {[B1(∆+t + 1)− B0]− [∂t − L]}um.

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  • 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

    Tmj → 0, as τ(h)→ 0, ∀mτ ≤ tmax, ∀j ∈ JΩ.

    In particular, the difference scheme is said to be consistent with the problem in the norm ‖ · ‖, if

    τ

    m−1∑ l=0

    ‖T l‖ → 0, as τ(h)→ 0, ∀mτ ≤ tmax.

    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

    |Tmj | ≤ C [τp + hq] , as τ(h)→ 0, ∀mτ ≤ tmax, ∀j ∈ JΩ,

    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

    τ

    m−1∑ l=0

    ‖T l‖ ≤ C [τp + hq] , as τ(h)→ 0, ∀mτ ≤ tmax.

    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 Um to the scheme satisfies

    ‖Um − um‖ → 0, as τ(h)→ 0, mτ → t ∈ [0, tmax],

    for all initial data u0 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

    ‖Um − um‖ ≤ C [τp + hq] , as τ(h)→ 0, mτ → t ∈ [0, tmax].

    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 K1 > 0 independent of τ , h, V

    0 and W 0, such that

    ‖V m −W m‖ ≤ K1‖V 0 −W 0‖, ∀mτ ≤ tmax,

    as long as V m −W m is a solution to the homogeneous difference scheme (meaning Fm = 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−11 B0(V m−1 −W m−1), thus, a scheme is (Lax-Richtmyer) stable if and only if

    ‖ ( B−11 B0

    )m ‖ ≤ K1, ∀mτ ≤ tmax. 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 u0 with which the corresponding problem is well posed, there exists an sequence of sufficiently smooth solutions vα, s.t. limα→∞ ‖v 0α − u0‖ → 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

    B1(U m+1 − um+1) = B0(Um − um)− Tm,

    or equivalently Um+1 − um+1 = (B−11 B0)(Um − um)− B −1 1 T

    m.

    Recursively, and by assuming U0 = u0, we obtain

    Um − um = − m−1∑ l=0

    (B−11 B0) lB−11 T

    m−l−1.

    Thus, by the uniform solvability ‖B−11 ‖ ≤ Kτ and the stability ‖(B−11 B0)l‖ ≤ K1, we have ‖Um − um‖ ≤ KK1τ

    ∑m−1 l=0 ‖T l‖, ∀m > 0.

    Therefore, by the definition of the consistency, we have

    lim τ(h)→0

    ‖Um − um‖ = 0, 0 ≤ mτ ≤ tmax.

  • Proof of Sufficiency of the Lax Equivalence Theorem (continue)

    For a general solution u, let vα be the smooth solution sequence satisfying limα→∞ ‖v 0α − u0‖ → 0.

    1 ∀ε > 0, ∃A > 0, such that ‖v 0α − u0‖ < ε for all α > A. 2 For fixed β > A, let V mβ be the solution of the difference

    scheme with V 0β = v 0 β .

    3 ∀ε > 0, ∃h(ε) > 0, s.t. ‖V mβ − vmβ ‖ < 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

    ‖Um − um‖ ≤ ‖Um − V mβ ‖+ ‖V mβ − vmβ ‖+ ‖vmβ − um‖ ≤ (K1 + 1 + C )ε.

    Since ε is arbitrary, this implies

    lim τ(h)→0

    ‖Um − um‖ = 0, 0 ≤ mτ ≤ tmax.

  • 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−11 B0) m h : bounded linear operators in (X, ‖ · ‖).

    2 (B−11 B0) m h � (B

    −1 1 B0)

    m̂ ĥ

    , either h < ĥ, or h = ĥ and m > m̂.

    3 By the resonance theorem, if, for each given u0 ∈ X ,

    Stmax(u 0) := sup

    {h>0, m≤τ−1tmax}

    { ‖(B−11 B0)

    m h u

    0‖ }

  • More on Consistency, Stability and Convergence

    Con