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In this question, you are asked to find the solution u(x, y) that satisfies the partial differential equation below using the spectral collocation method (with Gauss-Lobatto and also evenly spaced points), 8² u 8²u + მ2 дуг f(x, y). In order to do that, you will need to first find the derivative matrices and use them to construct the system of algebraic equation that looks something like [A] {u} = {f}. To extend the one-dimensional spectral collocation method with Gauss-Lobatto points into two dimensions, we can tile the one dimensional mesh along the second direction. Suppose that {X} and {Y} are arrays containing Nth order (N + 1 point) one-dimensional meshes, then we generate our two-dimensional mesh by repeating {X} at a spacing determined by the points in {Y}. The idea is akin to how MATLAB's meshgrid() function works. To illustrate this consider a two-dimensional mesh generated from the third order Gauss--Lobatto points.

In this question, you are asked to find the solution u(x, y) that satisfies the partial differential equation below using the spectral collocation method (with Gauss-Lobatto and also evenly spaced points), 8² u 8²u + მ2 дуг f(x, y). In order to do that, you will need to first find the derivative matrices and use them to construct the system of algebraic equation that looks something like [A] {u} = {f}. To extend the one-dimensional spectral collocation method with Gauss-Lobatto points into two dimensions, we can tile the one dimensional mesh along the second direction. Suppose that {X} and {Y} are arrays containing Nth order (N + 1 point) one-dimensional meshes, then we generate our two-dimensional mesh by repeating {X} at a spacing determined by the points in {Y}. The idea is akin to how MATLAB's meshgrid() function works. To illustrate this consider a two-dimensional mesh generated from the third order Gauss--Lobatto points.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Y 1 0.5 0 -0.5 −1 213,0 = 2413 212,0 219 = 211,0 = 215 −1 240,0 = 241 213,1 = 244 212,1 = 240 211,1 = 216 240,1 = 212 -0.5 u(x, y) ≈ Hence the second derivatives are given by 0 N N X 243,2 = 2415 i=0 j=0 212,2 = 2411 241,2 = 214 20,2 = 213 0.5 ΣΣυ.L;(2)L;(y). Here the nodes follow the lexicographic ordering from left to right, bottom to top. The advantage of this type of construction is that it essentially decouples the axes, meaning that we can take the one dimensional Lagrange basis functions and multiply them together to produce the two-dimensional interpolants 1 Uxx (x, y) _ouj, L{ (æ)Lj(y), Uyy (x, y) = U₁L₁(x) L (y). થરૂ,3 = 16 212,3 = 242 211,3 = 218 240,3 = 24
Transcribed Image Text:Y 1 0.5 0 -0.5 −1 213,0 = 2413 212,0 219 = 211,0 = 215 −1 240,0 = 241 213,1 = 244 212,1 = 240 211,1 = 216 240,1 = 212 -0.5 u(x, y) ≈ Hence the second derivatives are given by 0 N N X 243,2 = 2415 i=0 j=0 212,2 = 2411 241,2 = 214 20,2 = 213 0.5 ΣΣυ.L;(2)L;(y). Here the nodes follow the lexicographic ordering from left to right, bottom to top. The advantage of this type of construction is that it essentially decouples the axes, meaning that we can take the one dimensional Lagrange basis functions and multiply them together to produce the two-dimensional interpolants 1 Uxx (x, y) _ouj, L{ (æ)Lj(y), Uyy (x, y) = U₁L₁(x) L (y). થરૂ,3 = 16 212,3 = 242 211,3 = 218 240,3 = 24
In this question, you are asked to find the solution u(x, y) that satisfies the partial differential equation below using the spectral collocation method (with Gauss-Lobatto and also evenly spaced points), 8² u 8² u əx² მყ2 + = = f(x, y). In order to do that, you will need to first find the derivative matrices and use them to construct the system of algebraic equation that looks something like [A]{u} = {f}. To extend the one-dimensional spectral collocation method with Gauss-Lobatto points into two dimensions, we can tile the one dimensional mesh along the second direction. Suppose that {X} and {Y} are arrays containing Nth order (N + 1 point) one-dimensional meshes, then we generate our two-dimensional mesh by repeating {X} at a spacing determined by the points in {Y}. The idea is akin to how MATLAB's meshgrid() function works. To illustrate this consider a two-dimensional mesh generated from the third order Gauss--Lobatto points.
Transcribed Image Text:In this question, you are asked to find the solution u(x, y) that satisfies the partial differential equation below using the spectral collocation method (with Gauss-Lobatto and also evenly spaced points), 8² u 8² u əx² მყ2 + = = f(x, y). In order to do that, you will need to first find the derivative matrices and use them to construct the system of algebraic equation that looks something like [A]{u} = {f}. To extend the one-dimensional spectral collocation method with Gauss-Lobatto points into two dimensions, we can tile the one dimensional mesh along the second direction. Suppose that {X} and {Y} are arrays containing Nth order (N + 1 point) one-dimensional meshes, then we generate our two-dimensional mesh by repeating {X} at a spacing determined by the points in {Y}. The idea is akin to how MATLAB's meshgrid() function works. To illustrate this consider a two-dimensional mesh generated from the third order Gauss--Lobatto points.
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For convenience, we write the coefficients uj,¿ in the vector {u} = (u₁, U₂,. where l - 1 + i + (N + 1)j. Q5 Ulg... ... 9 Then the second derivative matrix may be written as where is some operation on the one-dimensional second derivative matrix [D²] and the identity matrix [I] that produces the correct two-dimensional derivative matrix. [D²].x Urr (xi, Yi) = [I] [D²]{u}, ® x x Uyy (Xi, Yi) = [D²] [I]{u}, ‚ U(N+1)²)™ = (u0,0, U0,1, . . . , Wi‚j: For the moment, assume a 2 × 2 mesh. Show that the two dimensional derivative matrices are given by [D²] y = = [A] [B] D₁₁ D12 D² D2 11 21 D₁₁ D²₁ D%2 The result above can be extended to an N x N mesh. The operation the Kronecker product which is implemented in MATLAB using the kron() matrix [A] and a p × q matrix [B], it is defined as D²₁ D²21 11 D²₁₁ D₁2 D21 D22 D112 D 222 a₁1[B] ⠀ am1[B] ‚UN,N)T D²12 ain [B] : amn [B] on two matrices is called function. For an m × n
Transcribed Image Text:For convenience, we write the coefficients uj,¿ in the vector {u} = (u₁, U₂,. where l - 1 + i + (N + 1)j. Q5 Ulg... ... 9 Then the second derivative matrix may be written as where is some operation on the one-dimensional second derivative matrix [D²] and the identity matrix [I] that produces the correct two-dimensional derivative matrix. [D²].x Urr (xi, Yi) = [I] [D²]{u}, ® x x Uyy (Xi, Yi) = [D²] [I]{u}, ‚ U(N+1)²)™ = (u0,0, U0,1, . . . , Wi‚j: For the moment, assume a 2 × 2 mesh. Show that the two dimensional derivative matrices are given by [D²] y = = [A] [B] D₁₁ D12 D² D2 11 21 D₁₁ D²₁ D%2 The result above can be extended to an N x N mesh. The operation the Kronecker product which is implemented in MATLAB using the kron() matrix [A] and a p × q matrix [B], it is defined as D²₁ D²21 11 D²₁₁ D₁2 D21 D22 D112 D 222 a₁1[B] ⠀ am1[B] ‚UN,N)T D²12 ain [B] : amn [B] on two matrices is called function. For an m × n
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