# sandbox/easystab/jacobian_formula.m

Validation: Comparing the expression of the analytical Jacobian and the numerical Jacobian for the functions on a mapped domain. We are only interested in the part of the Jacobian where the expression of the stretching plays a role. This is a follow-up of the code stretching_formula.m and to be used in domain_derivative_mapping.m.

disp('%%%%%%%')
clear all; clf;

% parameters
Nx=10;           % number of grid points in x
Ny=10;            % number of grid points in y
Lx=1;      % the length in x of the computational domain
Ly=1;       % the length in x of the computational domain
eps=1e-8;   % Small perturbation for computing the numerical Jacobian
method='cheb'; % discretization method
pts=5; % number of points in the stencil in case of finite differences

% differentiation
[d.x,d.xx,d.wx,x]=dif1D(method,0,Lx,Nx,pts);
[d.y,d.yy,d.wy,y]=dif1D(method,0,Ly,Ny,pts);
[D,l,X,Y,Z,I,NN]=dif2D(d,x,y);

% An operator to transform from the 1D x grid to the 2D grid
T=kron(speye(Nx,Nx),ones(Ny,1));
yy=Y(:);

l.phi=(1:NN)';
l.e=NN+[1:Nx]';
n=NN+Nx;

% base state
e=1+0.2*sin(pi*x/Lx);
phi=cos(pi*X).*cos(pi*Y);
q=[phi(:); e(:)];

% Computing Jacobian with respect to e
A=zeros(3*NN,Nx);
for gre=0:Nx

% Perturbation of one degree of freedom
if gre>0;
qq=q+((1:n)==(l.e(gre)))'*eps;
else
qq=q;
end

% computational domain derivatives
phi=qq(l.phi); phiy=D.y*phi;  phixx=D.xx*phi; phiyy=D.yy*phi; phixy=D.x*phiy;
e=qq(l.e); ex=d.x*e; exx=d.xx*e; E=T*e; Ex=T*ex; Exx=T*exx;


# The physical space differentiation matrices

Here instead of using the general function map2D.m to get the differentiation matrices on the deformed domain, we build them from the analytical mapping as derived and tested in stretching_formula.m#the-mapping.

    % Physical space diferentiation matrices
Dp.lap=D.xx ...
+spd(-2*yy.*E.^-1.*Ex)*(D.x*D.y) ...
+spd(yy.^2.*E.^-2.*Ex.^2+E.^-2)*D.yy ...
+spd(yy.*(2*E.^-2.*Ex.^2-E.^-1.*Exx))*D.y;
Dp.x=D.x+spd(-yy.*E.^-1.*Ex)*D.y;
Dp.y=spd(E.^-1)*D.y;

% Compute the physical space derivatives
phiplap=Dp.lap*phi;
phipx=Dp.x*phi;
phipy=Dp.y*phi;

% nonlinear function
feps=[phiplap; phipx; phipy];

% store Jacobian
if gre==0;

% The analytical expressions for the Jacobian
f=feps;


# The analytical expression of the Jacobian

When we just had to get the Jacobian of derivatives it was easy because these are linear operators, thus the Jacobian is the same as the operator itself. Now it is a bit more complicated because the expression of the derivative depends on the domain stretching \eta, and thus the derivatives are nonlinear functions of \eta1. In the following we denote \displaystyle \partial_\bar{x}^{\eta} the physical-space differentiation operator based on a domain stretched by \eta. And we are interested in how this thing depends on \eta for a small perturbation \tilde{\eta}.

# First derivative in y

We have \displaystyle \partial^\eta_\bar{y}\phi=\eta^{-1}\phi_y thus considering the perturbations of \phi and \eta we have \displaystyle \partial^{\eta+\tilde{\eta}}_\bar{y}(\phi+\tilde{\phi}) =(\eta+\tilde{\eta})^{-1}(\phi_y+\tilde{\phi}_y)\approx \eta^{-1}\phi_y+\eta^{-1}\partial_y\tilde{\phi}-\eta^{-2}\phi_y\tilde{\eta} which we can write in matrix form as \displaystyle \partial^{\eta+\tilde{\eta}}_\bar{y}(\phi+\tilde{\phi}) \approx \partial^\eta_\bar{y}\phi+ \begin{pmatrix} \partial^\eta_\bar{y} &-\eta^{-2}\phi_y\end{pmatrix} \begin{pmatrix} \tilde{\phi}\\ \tilde{\eta} \end{pmatrix}

# First derivative in x

We have \displaystyle \partial^\eta_\bar{x}\phi=\phi_x-y\eta^{-1}\eta_x\phi_y then considering the perturbations of \phi and \eta we have \displaystyle \partial^{\eta+\tilde{\eta}}_\bar{x}(\phi+\tilde{\phi}) \approx \partial^{\eta}_\bar{x}\phi+ \begin{pmatrix} \partial^\eta_\bar{x} &-y\phi_y(\eta^{-1}\partial_x-\eta^{-2}\eta_x) \end{pmatrix} \begin{pmatrix} \tilde{\phi}\\ \tilde{\eta} \end{pmatrix}

# For the laplacian

Now for the physical-space Laplacian \displaystyle \bar{\Delta}^{\eta}=\partial^\eta_{\bar{xx}}\partial^\eta_{\bar{yy}} we have \displaystyle \bar{\Delta}^{\eta+\tilde{\eta}}(\phi+\tilde{\phi}) =\bar{\Delta}_\eta \phi+ \begin{pmatrix} \bar{\Delta}^{\eta} & \phi_{xx}A+\phi_{yy}B+\phi_{xx}C \end{pmatrix} \begin{pmatrix} \tilde{\phi}\\ \tilde{\eta} \end{pmatrix} with A, B and C the Jacobians of the functions of \eta \displaystyle \begin{array}{l} a(\eta)=-2y\eta^{-1}\eta_x\\ b(\eta)=y^2\eta^{-2}\eta_x^2+\eta^{-2}\\ c(\eta)=y(2\eta^{-2}\eta_x^2-\eta^{-1}\eta_{xx} \end{array} After some algebra, we find \displaystyle \begin{array}{l} A=-2y(\eta^{-1}\partial_x-\eta^{-2}\eta_x)\\ B=2y^2\eta^{-2}(\eta_x\partial_x-\eta^{-1}\eta_x^2)-2\eta^{-3}\\ C=y[ 4\eta^{-2} (\eta_x\partial_x-\eta^{-1}\eta^2_{x})-\eta^{-1}(\partial_{xx}-\eta^{-1}\eta_{xx})] \end{array}

Now the difficulty with these expressions is that y is defined for everypoint of the grid, but \eta is only define on a 1D grid, so we need to use an operator that multiplies the size Nx vector of \eta to give the grid equivalent of \eta. This is the operator T built above.

        AA=-2*spd(yy)*T*(diag(e.^-1)*d.x-diag(e.^-2.*ex));
BB=2*(spd(yy.^2)*T*(diag(e.^-2.*ex)*d.x-diag(e.^-3.*ex.^2))-T*diag(e.^-3));
CC=spd(yy)*T*(diag(4*e.^-2.*ex)*d.x-diag(e.^-1)*d.xx+diag(-4*e.^-3.*ex.^2+e.^-2.*exx));

Dp.lapphie=spd(phixy)*AA+spd(phiyy)*BB+spd(phiy)*CC;
Dp.xphie=spd(yy.*phiy)*T*(diag(-e.^-1)*d.x+diag(e.^-2.*ex));
Dp.yphie=spd(phiy)*T*diag(-e.^-2);

else
% One column of the numerical Jacobian
A(:,gre)=(feps-f)/eps;
end
end

% Validation: compare numerical and analytical Jacobians
err_lap=norm(A(1:NN,:)-Dp.lapphie)
err_x=norm(A(NN+(1:NN),:)-Dp.xphie)
err_y=norm(A(2*NN+(1:NN),:)-Dp.yphie)

% Show the function phi and e
Yp=reshape(E.*Y(:),Ny,Nx);
mesh(X,Yp,reshape(phi,Ny,Nx));
hold on
plot(x,e,'k-',[0 Lx],[Ly Ly],'k--');
view(103,66);
xlabel('x');ylabel('y');zlabel('h');

set(gcf,'paperpositionmode','auto')
print('-dpng','-r80','jacobian_formula.png')


This gives the reassuring screen output

err_lap =
9.5728e-04
err_x =
9.9049e-06
err_y =
1.0995e-06

# The figure

The figure just shows the reference state \phi and \eta used to test the analytical expressions of the Jacobian. 