Matrix Stability Analysis

Consider the initial boundary value problem (IBVP)

ut = σuxx, 0 < x < 1, t > 0 (1)

u(0, t) = g(t), u(1, t) = h(t) (2)

u(x, 0) = f(x) (3)

Equation (1) can be written asut = Lu, (4)

where L is a linear differential operator.

We have seen three different numerical schemes to approximate the solution of IBVP(1)-(3). They are

1. Forward in time–Centered in space

Un+1i = rUn

i−1 + (1− 2r)Uni + rUn

i+1, i = 1, . . .m, (5)

where r = σ∆t/∆x2. This scheme is O(∆t)+O(∆x2). The linear system that resultsfrom (5) can be represented by

Un+1 = LF∆Un (6)

2. Backward in time–Centered in space

−rUn+1i−1 + (1 + 2r)Un+1

i − rUn+1i+1 = Un

i , i = 1, . . .m (7)

This scheme is O(∆t) + O(∆x2) The linear system that results from (7) can berepresented by

LB∆Un+1 = Un (8)

3. Crank–Nicholson

−r2Un+1i−1 + (1 + r)Un+1

i − r

2Un+1i+1 =

r

2Uni−1 + (1− r)Un

i +r

2Uni+1, i = 1, . . .m (9)

This scheme is O(∆t2) + O(∆x2) The linear system that results from (9) can berepresented by

LS∆Un+1 = LG

∆Un + Cn (10)

0.1 Definition 1: Stability of Linear Finite Difference Methods

A linear finite difference method (FDM) of the form

Un+1 = L∆Un (11)

corresponding to an IBVP of (4) (such as (1)-(3)) is stable if there exists C > 0, independentof the mesh spacing and the initial data, such that

||Un|| ≤ C||U0||, n→∞, ∆t→ 0, ∆x→ 0, n∆t ≤ T (12)

0.2 Theorem 1: Equivalent Condition

The FDM (11) is stable if and only if there exists a constant C > 0 independent of ∆xand ∆t such that

||(L∆)n|| ≤ C, n→∞, ∆t→ 0, ∆x→ 0, n∆t ≤ T (13)

Remark: Notice that C may be greater than 1.

Proof.Notice that

Un = L∆Un−1 = L∆

(L∆Un−2

)= L2

∆Un−2 = · · · = Ln∆U0

Therefore, for U0 6= 0

||Un|| ≤ C||U0|| ⇐⇒ ||Ln∆U0|| ≤ C||U0|| ⇐⇒ ||Ln

∆U0||||U0||

≤ C ⇐⇒ || (L∆)n || ≤ C

(14)

0.3 Corollary 1: Practical Condition

If the discrete operator L∆ of the FDM (11) satisfies

||L∆|| ≤ 1,

then the FDM (11) is stable.

Proof.Notice that ||Ln

∆|| ≤ ||L∆||n. Therefore, if

||L∆|| ≤ 1⇒ || (L∆)n || ≤ ||L∆||n ≤ 1

The stability follows from Theorem 1.

Remark: Apply this condition to the explicit FDM FT-CS using the infinity norm.

0.4 Corollary 2: More General Sufficient Condition

If there is is a c > 0 independent of ∆x and ∆t such that the discrete operator L∆ of theFDM (11) satisfies

||L∆|| ≤ 1 + c∆t,

for ∆t < ∆t∗, then the FDM (11) is stable.

Proof.Notice that n∆t ≤ T and 1 + c∆t ≤ ec∆t, then 1 + c∆t ≤ ecT/n. Therefore,

|| (L∆)n || ≤ ||L∆||n ≤ (1 + c∆t)n ≤ ecT = ec̃ = C

0.5 Definition 2: Spectral Radius

The spectral radius ρ(L∆) of the FDM matrix L∆ is the absolute value of its largesteigenvalue. Assuming that λi, i = 1, . . . N are the eigenvalues of L∆, then

ρ(L∆) = max1≤i≤N

|λi|

0.6 Theorem 2: Relationship Between Spectral Radius and Normof L∆

If ρ(L∆) and L∆ are the spectral radius and the vector-induced norm of L∆ then,

ρ(L∆) ≤ ||L∆||

Proof.For any eigenvector xi, it holds ||L∆xi|| = ||λixi||, for i = 1, 2, . . . , N . Therefore,

|λi| =||L∆xi||||xi||

≤ maxx 6=0||L∆x||||x||

≤ ||L∆|| ⇒ ρ(L∆) ≤ ||L∆||

0.7 Corollary 3: Necessary Condition

The conditionρn(L∆) ≤ C,

for a constant C > 0 independent of ∆x and ∆t is a necessary condition for the stabilityof the FDM (11).

Proof.Notice that ρn(L∆) = ρ ((L∆)n) ≤ || (L∆)n)||. Therefore, if ρn(L∆) is not bounded then|| (L∆)n)|| is also not bounded and the FDM is not stable.

0.8 Corollary 4: A More Practical Condition (special matrices)

If L∆ of the FDM (11) is symmetric or similar to a symmetric matrix, then

ρ(L∆) ≤ 1,

for any ∆x and ∆t, is also a sufficient condition for stability in the Euclidean norm.

Proof.If L∆ is a symmetric matrix then the Eucledian norm ||L∆||2 =

√ρ(L∆LT

∆) = ρ(L∆).Therefore,

ρ(L∆) ≤ 1⇒ ||L∆||2 ≤ 1

and the stability follows from Corollary 1.

Remark: Apply this condition to show stability of FT-CS and BT-CS FDM for IBVP(1)-(3) with homogeneous boundary conditions.

0.9 Definition 4: Convergence

A finite difference approximation Un converges to the solution un (the restriction of theexact solution u(x, tn) to the mesh) on 0 < t ≤ T in a particular vector norm if

||un −Un|| → 0, n→∞, ∆x→ 0, ∆t→ 0, n∆t ≤ T (15)

Why do we want to prove stability for FDM such as (11) ap-proximating certain PDE problems modelled by (4)? The answer tothis question is found in the next theorem

0.10 Theorem 3: Lax-Equivalence Theorem

A consistent linear FDM such as (11) is convergent if and only if it is stable.

In many problems of practical interest, we would like to study stability when t → ∞.To analyze stability for these problems, we need an alternative stability definition.

0.11 Definition 3: Absolute Stability

A FDM such as (11) is absolutely stable for a given mesh (of size ∆x and ∆t) if

||Un|| ≤ ||U0||, n > 0 (16)

0.12 Definition 4: Unconditional Stability

A FDM such as (11) is unconditionally stable if it is absolutely stable for all choices ofmesh spacing ∆x and ∆t.