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

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

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Page 1: Numerical Solutions to Partial Differential Equations...Numerical Solutions to Partial Di erential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University More

Numerical Solutions toPartial Differential Equations

Zhiping Li

LMAM and School of Mathematical SciencesPeking University

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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

∂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 withrespect to x , and is assumed not explicitly depend on the time t.

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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 thenorm ‖ · ‖ of a Banach space X, if it holds

1 for any given initial data u0 ∈ X, i.e. ‖u0‖ <∞, there existsa solution;

2 there is a constant C > 0, such that, if v , w are the solutionsof 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 theinitial-boundary value problems.

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Uniformly Well Conditioned Difference Schemes

We consider the difference scheme of the following form

B1Um+1 = B0Um + Fm.

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

2 (BαUm)j′ =∑

j∈JΩbαj′,jU

mj , ∀ j′ ∈ JΩ, α = 0, 1.

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

4 B1 = O(τ−1), B1 is invertible, and B−11 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 differencescheme as Um+1 = B−1

1 [B0Um + Fm] .

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

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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 := B1um+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, 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, iffor all sufficiently smooth exact solution u of the problem, thetruncation error satisfies

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

In particular, the difference scheme is said to be consistent withthe 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 andq 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 aconstant independent of j, τ and h.

Remark: Similarly, we can define the order of accuracy of a scheme withrespect to a norm ‖ · ‖, if

τ

m−1∑l=0

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

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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 thescheme 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 thenorm ‖ · ‖.

Remark: Similarly, we can define the order of convergence of a schemeto 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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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 anorm ‖ · ‖ and a given refinement path τ(h), if there exists aconstant 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 differencescheme (meaning Fm = 0 for all m) with initial data V 0 −W 0.

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Stability and Lax Equivalence Theorem

Stability and Uniform Well-Posedness of Finite Difference Schemes

The stability of finite difference schemes are closely related to theuniform 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−1

1 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 whichthe corresponding problem is well posed, there exists an sequenceof sufficiently smooth solutions vα, s.t. limα→∞ ‖v 0

α − u0‖ → 0.

Theorem

For a uniformly solvable linear finite difference scheme which isconsistent with a well-posed linear evolution problem, the stabilityis 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).

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Proof of Sufficiency of the Lax Equivalence Theorem (stability⇒ convergence)

If u ∈ X is a sufficiently smooth solution of the problem, then, bythe definition of the truncation error, we have

B1(Um+1 − um+1) = B0(Um − um)− Tm,

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

1 Tm.

Recursively, and by assuming U0 = u0, we obtain

Um − um = −m−1∑l=0

(B−11 B0)lB−1

1 Tm−l−1.

Thus, by the uniform solvability ‖B−11 ‖ ≤ Kτ and the stability

‖(B−11 B0)l‖ ≤ K1, we have ‖Um − um‖ ≤ KK1τ

∑m−1l=0 ‖T l‖, ∀m > 0.

Therefore, by the definition of the consistency, we have

limτ(h)→0

‖Um − um‖ = 0, 0 ≤ mτ ≤ tmax.

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Proof of Sufficiency of the Lax Equivalence Theorem (continue)

For a general solution u, let vα be the smooth solution sequencesatisfying limα→∞ ‖v 0

α − 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 − vm

β ‖ < ε, for all h < h(ε).

Thus, by the stability and the uniform invertibility of the schemeand the well-posedness of the problem that, if h < h(ε), then

‖Um − um‖ ≤ ‖Um − V mβ ‖+ ‖V m

β − vmβ ‖+ ‖vm

β − um‖ ≤ (K1 + 1 + C )ε.

Since ε is arbitrary, this implies

limτ(h)→0

‖Um − um‖ = 0, 0 ≤ mτ ≤ tmax.

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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)mh : bounded linear operators in (X, ‖ · ‖).

2 (B−11 B0)mh (B−1

1 B0)mh

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

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

Stmax(u0) := suph>0, m≤τ−1tmax

‖(B−1

1 B0)mh u0‖<∞,

then the sequence (B−11 B0)mh is uniformly bounded, and

consequently the scheme is stable.

Suppose for some u0 ∈ X, ∃τk → 0, mkτk → t ∈ [0, tmax], s.t.

limk→∞

‖(B−11 B0)mk

hku0‖ =∞.

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Consistency, Stability and Convergence of FDM

Stability and Lax Equivalence Theorem

Proof of Necessity of the Lax Equivalence Theorem (continue)

4 u, U with initial data u0, w and W with initial data w 0 = 0.

(B−11 B0)mh u0 = Um

h −W mh = (Um

h −um)+(um−wm)+(wm−W mh ).

5 Convergence ⇒limk→∞

(‖Umk

hk− umk‖+ ‖wmk −W mk

hk‖)

= 0.

6 Well-posedness ⇒ ‖um − wm‖ ≤ C‖u0 − w 0‖ = C‖u0‖.

7 Therefore limk→∞

‖(B−11 B0)mk

hku0‖ ≤ C‖u0‖. (A contradiction).

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In What Norm Should the Stability be Analyzed

How to Choose a Norm for Stability Analysis

1 Maximum principle ⇒ ‖ · ‖∞ a good candidate.

2 Discrete maximum principle ⇒ ‖ · ‖∞ stability.

3 Constant-coefficient equation, periodic boundary conditions⇒ ‖ · ‖2 a good choice.

4 Fourier analysis method ⇒ ‖ · ‖2 stability conditions.

5 Local Fourier analysis ⇒ necessary ‖ · ‖2 stability conditions.

6 More general cases ⇒ ‖ · ‖1, ‖ · ‖p, energy norm, e.t.c..

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In What Norm Should the Stability be Analyzed

Small Disturbance on a Scheme Does Not Change Its Stability

Theorem

Suppose Um+1 = B−11 B0Um is stable in ‖ · ‖. Suppose E (τ) is

uniformly bounded in ‖ · ‖. Then, the scheme

Um+1 =[B−1

1 B0 + τE (τ)]

Um

is also stable in the norm ‖ · ‖.

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In What Norm Should the Stability be Analyzed

Small Disturbance on a Scheme Does Not Change Its Stability

Proof:

(A + B)2 = A2 + AB + BA + B2, in general (A + B)m has 2m

terms, there are Cmj terms having j of B and (m − j) of A

multiplied together in various order.

‖(B−11 B0)k‖ ≤ K1, for all k, and ‖E (τ)‖ ≤ K2.

Therefore, if mτ ≤ tmax, we have

‖[B−1

1 B0 + τE (τ)]m ‖ ≤ m∑

j=0

Cmj K j+1

1 (τK2)j

= K1(1 + K1K2τ)m ≤ K1etmaxK1K2 .

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Fourier Analysis and von Neumann Condition of L2 Stability

Fourier Analysis in Constant-Coefficient and Periodic Case (Ω = (−1, 1)n)

1 Coefficients of a scheme: bαj′,j = bα(j−j′) (see (4.1.3)).

2 Fourier mode for a one-step scalar scheme: Umj = λmk e iπk·jh,

amplification factor: λk =

∑j b0

(j−j′)e iπk·(j−j′)h∑

j b1(j−j′)e iπk·(j−j′)h

=B0(k)

B1(k).

3 In vector cases, apply the discrete Fourier transform:

U(k) =1

(√

2N)n

N∑j1=−N+1

· · ·N∑

jn=−N+1

Uje−iπk·j 1

N ,

discrete inverse Fourier transform (i.e. Fourier expansion):

Uj =1

(√

2)n

N∑k1=−N+1

· · ·N∑

kn=−N+1

U(k)e iπk·j1N .

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Fourier Analysis and von Neumann Condition of L2 Stability

Fourier Analysis in Vector Case (Constant-Coefficient and Periodic)

4 To obtain the characteristic equations, substitute the Fourier

mode Umj (k) = Um(k)e iπk·j

1N into the homogeneous scheme∑

j b1j′,jU

m+1(k)e iπk·j1N =

∑j b0

j′,jUm(k)e iπk·j

1N .

5 Denote Bα =∑

j bαj′,jeiπk·(j−j′) 1

N =∑

j bα(j−j′)e iπk·(j−j′) 1

N .

6 We have B1Um+1(k) = B0Um(k).

7 Characteristic equations in the space of frequencies:

Um+1(k) = B1(k)−1B0(k)Um(k),

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Fourier Analysis and von Neumann Condition of L2 Stability

Fourier Analysis in Vector Case (Constant-Coefficient and Periodic)

9 Amplification matrix: G (k) = B1(k)−1B0(k) ∈ Cp×p,

10 Fourier mode: Umj = λmk e iπk·jhU(k) with λkU(k) = G (k)U(k).

11 If G (k) is a normal matrix, i.e. GTG = G GT , then G has porthogonal eigenvectors. Therefore ‖Um‖2 = |λk|m‖U0‖2.

12 In the general case, Um(k) = [G (k)]m U0(k), thereforeL2 stability ⇔ ‖ [G (k)]m ‖2 ≤ K1, ∀ k, ∀mτ ≤ tmax.

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Fourier Analysis and von Neumann Condition of L2 Stability

The von Neumann Condition of L2 Stability

Theorem

A necessary condition for the difference scheme to be L2 stablealong a refinement path is that there exists a constant K ≥ 0 suchthat every eigenvalue λk of the amplification matrix G (k) satisfies

|λk| ≤ 1 + Kτ, ∀ τ ≤ tmax, ∀ k.

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Fourier Analysis and von Neumann Condition of L2 Stability

The von Neumann Condition of L2 Stability

Proof:

1 L2 stable ⇒ for |λk|m ≤ K1 for m = [tmax/τ ].

2 Let f (x) = x s , (s < 1), since f ′′(x) = s(s − 1)x s−2 < 0, f (x)is a concave function of x , hence f (x) ≤ f (1) + f ′(1)(x − 1).

3 K1 ≤ 1 ⇒ |λk| ≤ 1. K1 > 1 ⇒ |λk| ≤ K1m

1 ≤ 1 + 1m (K1 − 1).

4 If 2τ ≤ tmax, m ≥ tmax

τ− 1 ≥ tmax

2τ.

5 The theorem holds for K = 2(K1 − 1)/tmax.

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Fourier Analysis and von Neumann Condition of L2 Stability

In General, the von Neumann Condition Is Not Sufficient for Stability

1 If G (k) is a normal matrix, then the von Neumann conditionis also a sufficient condition for a scheme to be L2 stable.

2 The conclusion is however not valid if G (k) is not normal.

3 In general, we can not expect to obtain a sufficient conditionby the local Fourier analysis (see Exercise 4.3).

4 In practical computations, the constant K can be a problem.If K is too big, then the von Neumann condition can fail toprovide useful information on the grid ratio.

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Strong Stability and Practical Stability Conditions

More Practical Conditions Than the von Neumann Condition Is Necessary

1 Initial value problem of ut + aux = cuxx , (c > 0).

2 Explicit central difference scheme:

Um+1j − Um

j

τ+ a

Umj+1 − Um

j−1

2h= c

Umj+1 − 2Um

j + Umj−1

h2.

3 |λk |2 = [1− 2µ(1− cos kh)]2 + [ν sin kh]2.

4 kh = π ⇒ µ ≤ 12 a necessary condition for the L2 stability.

5 µ ≤ 12 ⇒ [1− 2µ(1− cos kh)]2 ≤ 1, ν2 = a2

c µτ ≤a2

2c τ

⇒ |λk |2 ≤ 1 + a2

2c τ ⇒ the von Neumann condition is satisfied.

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Strong Stability and Practical Stability Conditions

More Practical Conditions Than the von Neumann Condition Is Necessary

6 In fact, µ ≤ 12 ⇔ L2 stability.

7 Recall |λk |2 = [1− 2µ(1− cos kh)]2 + [ν sin kh]2.

8 Take µ = 14 , ν ≥ 1 (Peclet number ah

c = νµ ≥ 4),

9 then, for kh = π/2, the amplification factor satisfies |λk | ≥ 54 ,

hence, the corresponding Fourier mode grows dramatically.

10 Yet, all Fourier mode solutions to the original problem decays.(dissipation speed: e−ck2

)

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Strong Stability and Practical Stability Conditions

A Practical Stability Condition for Difference Schemes

Definition

Let u(k, t) be the Fourier mode solutions to the differentialequation, denote

α = infβ ≥ 0 : |u(k, t + τ)| ≤ eβτ |u(k, t)|, ∀ k.

If all of the amplification factors of the discrete Fourier modesolutions to the difference scheme satisfy

|λk| ≤ eατ , ∀ k,

then, the difference scheme is said to be practically stable, or tohave practical stability. In particular, if α = 0, the correspondingdifference scheme is said to be strongly stable, or to have strongstability.

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Strong Stability and Practical Stability Conditions

A Practical Stability Condition for Difference Schemes

If α > 0,

the discrete Fourier mode solutions can grow exponentiallyfast;

however, the growth speed ≤ the fastest growth speed of realFourier mode solution;

the discrete Fourier modes are ”relatively” not in growth.

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Strong Stability and Practical Stability Conditions

A Practical Stability Condition for Difference Schemes

If α = 0, the strong stability requires |λk| ≤ 1. In the aboveexample, |λk |2 = [1− 2µ(1− cos kh)]2 + [ν sin kh]2.

|λk | ≤ 1, ∀k ⇔ µ ≤ 12 and τ ≤ 2c

a2 (see (3.4.10)).

Since ν2 =(aτh

)2= a2

c µτ , τ ≤ 2ca2 ⇔ ν2 ≤ 2µ.

Take µ = 1/4, ν ≤ 1/√

2 ⇒ |λk |2 ≤ cos2 12 kh ≤ 1, ∀k .

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SK 4µ1, 2, 4

Thank You!