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Finite Element Analysis Toolbox
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ex_custom_equation1.m File Reference

Description

EX_CUSTOM_EQUATION1 1D Black-Scholes custom equation example.

[ FEA, OUT ] = EX_CUSTOM_EQUATION1( VARARGIN ) 1D Black-Scholes model equation example using the custom equation physics mode. Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
icase       scalar {1}/2           Test case equation to solve
tmax        scalar {1}             Maximum/stopping time
len         scalar {1}             Length of domain
hmax        scalar {1/20}          Grid cell size
ischeme     scalar {3}             Time stepping scheme
sfun        string {sflag1}        Finite element shape function
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output stuct

Code listing

 cOptDef = { ...
   'icase',    1; ...
   'tmax',     1; ...
   'len',      1; ...
   'hmax',     1/20; ...
   'ischeme'   3; ...
   'sfun',     'sflag1'; ...
   'iplot',    1; ...
   'dvname',   'u'; ...
   'tol',      1.1e-2; ...
   'fid',      1 };
 [got,opt] = parseopt( cOptDef, varargin{:} );
 fid       = opt.fid;

% Grid generation.
 nx       = round( opt.len/opt.hmax );
 fea.grid = linegrid( nx, 0, opt.len );


% Problem definition.
 u = opt.dvname;
 switch( opt.icase )
   case 1
     seqn = [u,''' - 1/2*',u,'x_x - ',u,'x_t + ',u,'_t = (x-t)^5 - 10*(x-t)^4 - 10*(x-t)^3'];
   case 2
     seqn = [u,''' - 1/2*x^2*',u,'x_x - x*',u,'x_t + ',u,'_t = (x-t)^5 - 5*(x-t)^4 - 5*x*(x-t)^4 - 10*x^2*(x-t)^3'];
 end
 refsol = '(x-t).^5';
 init_u = 'x^5';


% Set up problem struct.
 fea.sdim  = { 'x' };
 fea = addphys( fea, @customeqn );
 fea.phys.ce.dvar = { u };
 fea.phys.ce.eqn.seqn = seqn;
 fea.phys.ce.sfun = { opt.sfun };


% Dirichlet BCs for the left and right boundaries.
 fea.phys.ce.bdr.coef{1,5} = { 1 1 };
 fea.phys.ce.bdr.coef{1,7} = { '-t^5' ['(',num2str(opt.len),'-t)^5'] };


% Check and parse problem struct.
 fea = parsephys( fea );
 fea = parseprob( fea );


% Call to time-dependent solver.
 [fea.sol.u,tlist] = solvetime( fea, 'fid',     fid, ...
                                     'tmax',    opt.tmax, ...
                                     'init',    init_u, ...
                                     'icub',    6, ...
                                     'ischeme', opt.ischeme, ...
                                     'tstep',   opt.tmax/100);

% Postprocessing.
 fea.sol.t = tlist(end);
 fea.sol.u = fea.sol.u(:,end);
 refsol    = strrep( refsol, 't', num2str(fea.sol.t) );
 if( opt.iplot>0 )

   figure
   postplot( fea, 'surfexpr', u, 'axequal', 'off', 'linewidth', 2 )
   title( ['Solution at time ',num2str(fea.sol.t)] )
   hold on
   x = linspace(0,opt.len,25);
   u_ref = eval(refsol);
   plot( x, u_ref, 'r--' )
 end


% Error checking.
 err = evalexpr( ['abs(',refsol,'-',u,')'], linspace(0,opt.len,10), fea );
 err = norm(err);

 if( ~isempty(fid) )
   fprintf(fid,'\nL2 Error: %f\n',err)
   fprintf(fid,'\n\n')
 end

 out.err  = err;
 out.pass = out.err<opt.tol;
 if ( nargout==0 )
   clear fea out
 end