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.

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 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.

3 / 30

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.

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

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