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The Spectral Collocation Method for Solving Boundary Value Problems In this livescript, you will learn how To solve boundary value problems using the spectral collocation method. Consider the linear two point boundary value problem \frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}-6y=0 with the boundary conditions y(-1)=1 an y'(1)=1 . y0 = 1;
yN = 1; To solve this numerically using the spectral collocation method, the steps are basically the same as with finite difference methods. Fortunately, meshing in the Spectral collocation method isn’t much of an issue as the points are defined using the Gauss-Lobatto nodes. N = 10;
xp = GLL_nodes(N); To solve the differential equation numerically, we first need to approximate the derivative terms, which can be done using the derivative matrix. D = DerivMatrix(xp,N);
D2 = D*D; Now, notice how we may write the differential equation as L[y]=\Big(\frac{d^{2}}{dx^{2}}+2\frac{d}{dx}-6\Big)y=0 This therefore shows us that we can approximate L as L = D2+2*D-6*eye(size(D));
b = zeros(N+1,1); Now, to apply boundary conditions, the Dirichlet boundary condition at x=-1 may be determined in the same manner as a finite difference method. By eliminating the corresponding rows and columns from the [L] matrix and moving the influence of the nodes to the \{b\} vector. L_sys = L(2:end,2:end);
b = b(2:end)-y0*L(2:end,1); As for the Neumann boundary condition, in this case we can’t just use a different differencing method near the wall, so we’ll have to modify the methods discussed earlier to use a spectral method. Since the derivatives are given by the derivative matrix, we replace the equation at the node x=1 with an equation for the derivative. \pmatrix{
D_{N0} & D_{N1} & \cdots & D_{NN}
}
\pmatrix{
y_{0}\cr
y_{1}\cr
\vdots\cr
y_{N}
}
=
y_{N} Expanding the y_{0} term out as it’s the dirichlet boundary condition at x=-1 , we have \pmatrix{
D_{N1} & \cdots & D_{NN}
}
\pmatrix{
y_{1}\cr
\vdots\cr
y_{N}
}
=
y_{N}
–
D_{N0}y_{0} which we may replace in [L] and \{b\} using L_sys(end,:) = D(end,2:end);
b(end) = yN-D(end,1)*y0; And so, solving gives sol = L_sys\b;
y = [y0;sol];
plot(linspace(-1,1,100),LagrangePolynomial(xp,y,N,linspace(-1,1,100)));
xlabel(‘x’)
ylabel(‘y’) function D=DerivMatrix(x,n)
D=zeros(n+1,n+1);
for i=1:n+1
num(i)=1.0;
for k=1:n+1
if k ~= i
num(i)=num(i)*(x(i)-x(k));
end
end
end
for j=1:n+1
den(j)=1.0;
for k=1:n+1
if k ~= j
den(j)=den(j)*(x(j)-x(k));
end
end
end
for i=1:n+1
for j=1:n+1
if i ~= j
D(i,j)=num(i)/(den(j)*(x(i)-x(j)));
end
end
end
for i=1:n+1
for k=1:n+1
if i~=k
D(i,i)=D(i,i)+1./(x(i)-x(k));
end
end
end
end
function yint=LagrangePolynomial(x,y,n,xint)
yint=zeros(size(xint));
for i=1:n+1
Lix=1.0;
for j=1:n+1
if j ~= i
Lix=Lix.*(xint-x(j))/(x(i)-x(j));
end
end
yint=yint+y(i).*Lix;
end
end
function xp = GLL_nodes(N)
syms x
Lo = (1-x^2)*diff(legendreP(N,x),x);
xp = double(vpasolve(Lo == 0));
end
manual code ready 0.4 figure fec59d85-4565-4271-8979-998b756599fa 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 577 433 14 15 16 14 16 17 18 true false 0 8 8 false false 1 9 9 false false 2 13 13 false false 3 14 14 false false 4 16 16 false false 5 17 17 false false 6 21 21 false false 7 22 22 false false 8 25 25 false false 9 26 26 false false 10 34 34 false false 11 35 35 false false 12 37 37 false false 13 38 38 false false 14 40 40 0 false false 15 41 41 0 false true 16 42 42 0
2020-09-26T05:07:59Z 2020-09-27T03:02:11Z
application/vnd.mathworks.matlab.code MATLAB Code R2020b
9.9.0.1444674 356bfa8a-ed9d-4820-96fb-ee77672b9bd7
9.9.0.1467703
R2020b
Aug 26 2020
2314982500