<?xml version="1.0" encoding="UTF-8"?> <simulation xmds-version="2"> <name>bessel_transform</name> <author>Graham Dennis</author> <description> Solve the wave equation on a sphere of radius 1 utilising spherical symmetry by using the Spherical Bessel function transform. </description> <features> <benchmark /> <!-- <error_check /> --> <bing /> <globals> <![CDATA[ const real T = 10.0; const real mass = 1e-3; const real length = 1.0; const real mu = mass/length; ]]> </globals> </features> <geometry> <propagation_dimension> t </propagation_dimension> <transverse_dimensions> <!-- Volume prefactor = 4\pi is to cover the suppressed integration over \theta and \phi --> <dimension name="x" lattice="100" spectral_lattice="50" domain="(0, 1)" transform="spherical-bessel" volume_prefactor="4.0*M_PI"/> </transverse_dimensions> </geometry> <vector name="main" initial_basis="x" type="complex"> <components> u uDot </components> <initialisation> <![CDATA[ u = exp(-100.0*(x-0.25)*(x-0.25)); uDot = 0.0; ]]> </initialisation> </vector> <sequence> <integrate algorithm="ARK45" tolerance="1e-6" interval="4e-3" steps="400"> <samples>100 100</samples> <operators> <operator kind="ex" basis="kx" type="real"> <operator_names>L</operator_names> <![CDATA[ L = -T*kx*kx/mu; ]]> </operator> <integration_vectors>main</integration_vectors> <![CDATA[ du_dt = uDot; duDot_dt = L[u]; ]]> </operators> </integrate> </sequence> <output> <sampling_group basis="x" initial_sample="yes"> <moments>amp</moments> <dependencies>main</dependencies> <![CDATA[ amp = u.Re(); ]]> </sampling_group> <sampling_group basis="kx" initial_sample="no"> <moments>amp</moments> <dependencies>main</dependencies> <![CDATA[ amp = u.Re(); ]]> </sampling_group> </output> </simulation>