Numerical Solutions to Partial Differential ... Numerical Solutions to Partial Di erential Equations...
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Numerical Solutions to Partial Differential Equations
Zhiping Li
LMAM and School of Mathematical Sciences Peking University
More on Consistency, Stability and Convergence
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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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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.
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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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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.
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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.
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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].
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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.
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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.
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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.
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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‖ }
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