Sets up shock at angle to grid to test 3D algorithms. More...

Functions
static void	lx_bc (GridS *pG)
	Apply boundary condition in left-x direction.
static void	rx_bc (GridS *pG)
	Apply boundary condition in right-x direction.
static void	ly_bc (GridS *pG)
	Apply boundary condition in left-y direction.
static void	ry_bc (GridS *pG)
	Apply boundary condition in right-y direction.
static void	lz_bc (GridS *pG)
	Apply boundary condition in left-z direction.
static void	rz_bc (GridS *pG)
	Apply boundary condition in right-z direction.
static Real	Ax (const Real x1, const Real x2, const Real x3)
	s-component of vector potential
static Real	Ay (const Real x1, const Real x2, const Real x3)
	y-component of vector potential
static Real	Az (const Real x1, const Real x2, const Real x3)
	z-component of vector potential
void	problem (DomainS *pDomain)
	Problem initial conditions, edited for each problem.
void	problem_write_restart (MeshS pM, FILE fp)
	Problem specific subroutine for use with writing restart files, Edited for each problem.
void	problem_read_restart (MeshS pM, FILE fp)
	Problem specific subroutine for use with reading restart files, Edited for each problem.
ConsFun_t	get_usr_expr (const char *expr)
	Sets pointer to expression for special output data.
VOutFun_t	get_usr_out_fun (const char *name)
	Returns a user defined output function pointer.
void	Userwork_in_loop (MeshS *pM)
	Problem work executed during each step in the main Athena loop, see main.c.
void	Userwork_after_loop (MeshS *pM)
	Problem work executed after the main Athena loop, see main.c Edited for each problem.
Variables
static int	rx
static int	ry
static int	rz
static Real	ang_2
static Real	ang_3
static Real	sin_a2
static Real	cos_a2
static Real	sin_a3
static Real	cos_a3
static Real	dl
static Real	vxl
static Real	vyl
static Real	vzl
static Real	dr
static Real	vxr
static Real	vyr
static Real	vzr
static Real	Bxl
static Real	Byl
static Real	Bzl
static Real	Bxr
static Real	Byr
static Real	Bzr
static Real	Pl
static Real	Pr

Detailed Description

Sets up shock at angle to grid to test 3D algorithms.

PURPOSE: Sets up shock at angle to grid to test 3D algorithms. Setting up the initial conditions to minimize grid noise is very complex, so this special function has been written to handle only 3D problems.

This problem cannot be run with Static Mesh Refinement.

This code is most easily understood in terms of a one dimensional problem in the coordinate system (x1,x2,x3). Two coordinate rotations are applied to obtain a new wave vector in a 3D space in the (x,y,z) coordinate system.

First rotate about the x2 axis:

x1' = x1*cos(ang_2) - x3*sin(ang_2)
x2' = x2
x3' = x1*sin(ang_2) + x3*cos(ang_2)

Next rotate about the x3' axis:

x = x1'*cos(ang_3) - x2'*sin(ang_3)
y = x1'*sin(ang_3) + x2'*cos(ang_3)
z = x3'

Expanding this out we get:

x = x1*cos(ang_2)*cos(ang_3) - x2*sin(ang_3) - x3*sin(ang_2)*cos(ang_3)
y = x1*cos(ang_2)*sin(ang_3) + x2*cos(ang_3) - x3*sin(ang_2)*sin(ang_3)
z = x1*sin(ang_2) + x3*cos(ang_2)

This inverts to:

x1 = x*cos(ang_2)*cos(ang_3) + y*cos(ang_2)*sin(ang_3) + z*sin(ang_2)
x2 = -x*sin(ang_3) + y*cos(ang_3)
x3 = -x*sin(ang_2)*cos(ang_3) - y*sin(ang_2)*sin(ang_3) + z*cos(ang_2) Note these transformations are the same used in linear_wave3d and cpaw3d

The initial conditions must be translation invariant, i.e. q(x,y,z) = q(x+tx, y+ty, z+tz) for (tx,ty,tz) such that x1(x,y,z) = x1(x+tx, y+ty, z+ty). Inserting this last expression for x1(x,y,z) into the transformation above leads to

tx*cos(ang_2)*cos(ang_3) + ty*cos(ang_2)*sin(ang_3) + tz*sin(ang_2) = 0.

Now, restrict the translation symmetry to be discrete by inserting (tx, ty, tz) = (nx*dx, ny*dy, nz*dz) where (nx, ny, nz) are integers (not the number of grid cells in each direction) and (dx, dy, dz) are the grid cell sizes. With some simplification the translation symmetry becomes

nx + ny*tan(ang_3)*(dy/dx) + nz*(tan(ang_2)/cos(ang_3))*(dz/dx) = 0.

Now, choose an integer unit cell size (rx, ry, rz) where

tan(ang_3)*(dy/dx) = (rx/ry)

and

(tan(ang_2)/cos(ang_3))*(dz/dx) = (rx/rz).

With this choice, or equation for discrete translation symmetry becomes

(nx/rx) + (ny/ry) + (nz/rz) = 0.

PRIVATE FUNCTION PROTOTYPES:

lx_bc() - use periodicity of "unit cell" to apply BCs to left-x edge
rx_bc() - use periodicity of "unit cell" to apply BCs to right-x edge
ly_bc() - use periodicity of "unit cell" to apply BCs to left-y edge
ry_bc() - use periodicity of "unit cell" to apply BCs to right-y edge
lz_bc() - use periodicity of "unit cell" to apply BCs to left-z edge
rz_bc() - use periodicity of "unit cell" to apply BCs to right-z edge
Ax() - x-component of vector potential for initial conditions
Ay() - y-component of vector potential for initial conditions
Az() - z-component of vector potential for initial conditions

This is the equation for a set of discrete points which lie in a plane and is a key relation for setting ghost cells in the boundary condition routines below. -- T. A. Gardiner -- 7/21/2006

Definition in file shkset3d.c.