Matrix Stability AnalysisConsider 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 are1. Forward in time–Centered in spaceUn+1i= rUni−1+ (1 − 2r)Uni+ rUni+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 byUn+1= LF∆Un(6)2. Backward in time–Centered in space−rUn+1i−1+ (1 + 2r)Un+1i− rUn+1i+1= Uni, i = 1, . . . m (7)This scheme is O(∆t) + O(∆x2) The linear system that results from (7) can berepresented byLB∆Un+1= Unor Un+1= (LB∆)−1Un(8)3. Crank–Nicholson−r2Un+1i−1+ (1 + r)Un+1i−r2Un+1i+1=r2Uni−1+ (1 − r)Uni+r2Uni+1, i = 1, . . . m (9)This scheme is O(∆t2) + O(∆x2) The linear system that results from (9) can berepresented byLS∆Un+1= LG∆Unor Un+1= (LS∆)−1LG∆Un= (LCN∆)Un(10)0.1 Definition 1: StabilityA linear finite difference method (FDM) of the formUn+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 ConditionThe 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 1.0.3 Corollary 1: Practical ConditionIf the discrete operator L∆of the FDM (11) satisfies||L∆|| ≤ 1,then the FDM (11) is stable.Remark: Apply this condition to the explicit FDM FT-CS using the infinity norm.0.4 Corollary 2: More General ConditionIf 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,then the FDM (11) is stable.0.5 Definition 2: Spectral RadiusThe 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∆, thenρ(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∆||0.7 Corollary 3: Necessary ConditionThe conditionρn(L∆) ≤ C,for a constant C > 0 independent of ∆x and ∆t is a necessary condition for the stabilityof the FDM (11).0.8 Corollary 4: A More Practical Condition (special matrices)If L∆of the FDM (11) is symmetric or similar to a symmetric matrix, thenρ(L∆) ≤ 1,for any ∆x and ∆t, is also a sufficient condition for stability in the Euclidean norm.Remark: Apply this condition to show stability of FT-CS and BT-CS FDM for IBVP(1)-(3) with homogeneous boundary conditions.Why do we want to prove stability for FDM such as (11) ap-proximating certain PDE problems modelled by (4)?The answer to this question is found in the next theorem0.9 Theorem 3: Lax-Equivalence TheoremA 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.0.10 Definition 3: Absolute StabilityA FDM such as (11) is absolutely stable for a given mesh (of size ∆x and ∆t) if||Un|| ≤ ||U0||, n > 0 (14)0.11 Definition 4: Unconditional StabilityA FDM such as (11) is unconditionally stable if it is absolutely stable for all choices ofmesh spacing ∆x and ∆t.Backward Time Centered in Space
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