EDGƎ

Equations

• advection (1D):

  Solves the one-dimensional advection equation. q(x,t) is a scalar. The scalar advection speed a(x) can be set per element, but has to be either positive or negative for the entire domain.

  q_t + a * q_x = 0
  

• advection (2D):

  Solves the two-dimensional advection equation. q(x,y,t) is a scalar. The scalar advection speeds a(x,y) and b(x,y) can be set per element. Each has to be either positive or negative for the entire domain.

  q_t + a * q_x + b * q_y = 0
  

• advection (3D):

  Solves the three-dimensional advection equation. q(x,y,z,t) is a scalar. The scalar advection speeds a(x,y,z), b(x,y,z) and c(x,y,z) can be set per element. Each has to be either positive or negative for the entire doman.

  q_t + a * q_x + b * q_y + c q_z = 0
  

• elastics (2D):

  Solves the two-dimensional elastic wave equations. The vector of quantities q(x,y,t)=(sigma_xx, sigma_yy, sigma_xy, u, v ) contains the normal stress components sigma_xx and sigma_yy, the shear stress sigma_xy and the two particle velocities u and v in x- and y-direction respectively. The Jacobians A(x,y) and B(x,y) are allowed to be set per element and summarize the material parameters.

  q_t + A q_x + B q_y = 0
  

• elastics (3D):

  Solves the three-dimensional elastic wave equations. The vector of quantities q(x,y,z,t)=(sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_xz, sigma_yz, u, v, w ) contains the normal stress components sigma_xx, sigma_yy and sigma_zz, the shear stresses sigma_xy, sigma_xz and sigma_yz and the three particle velocities u, v w in x-, y- and z-direction respectively. The Jacobians A(x,y,z), B(x,y,z) and C(x,y,z) are allowed to be set per element and summarize the material parameters.

  q_t + A q_x + B q_y + C q_z = 0
  

• swe (1D):

  Solves the one-dimensional Shallow Water Equations (SWE) in conservative form. The conserved quantities q(x,t)=(h,hu) are the water height h and the momentum hu. The flux function is nonlinear. Bathymetry is supported.

  q_t + f(q)_x = 0,
  
         |         hu           |
  f(q) = |                      |
         | hu^2 + 1/2 * g * h^2 |
  

• swe (2D):

  Solves the two-dimensional Shallow Water Equations (SWE) in conservative form. The conserved quantities q(x,t)=(h,hu,hv) are the water height h, the momentum hu in x-direction and the momentum hv in y-direction. The flux function is nonlinear. Bathymetry is supported.

  q_t + f(q)_x + g(q)_y = 0,
  
         |         hu           |         |          hv          |
         |                      |         |                      |
  f(q) = | hu^2 + 1/2 * g * h^2 |, g(q) = |          huv         |
         |                      |         |                      |
         |         huv          |         | hv^2 + 1/2 * g * h^2 |
  

Elements

• line (1D):

  Line element. Element width dx is allowed to change in every element.

• quad4r (2D):

  Rectangular, 4-node quadrilaterals. Widths dx and dy are allowed to change on a per-row/per-column basis (conforming mesh).

• tria3 (2D):

  3-node triangles. Arbitrary, conforming triangulations of the computational domain are supported.

• hex8r (3D):

  Rectangular, 8-node hexahedrons (bricks). Widths dx, dy and dz are allowed to change on a conforming mesh basis.

• tet4 (3D):

  4-node tetrahedrons. Arbitrary, conforming tetrahedralization are allowed.

Feature table

Based on the equations and the element type, the following table shows the implemented features:

equations	element types	CFR	FV	ADER-DG	LIBXSMM
advection	line, quad4r, tria3, hex8r, tet4	x	x	x
elastics	quad4r, tria3, hex8r, tet4	x	x	x	x
swe	line, quad4r, tria3	x	x

High Performance Support

Microarchitecture	Machine(s)
Sandy Bridge	Stampede 1
Bulldozer	Blue Waters
Haswell	Comet, Cori Phase 1
Knights Landing	Stampede 2, Cori Phase 2, Theta
Skylake	Amazon Elastic Compute Cloud, Google Cloud Platform, Stampede 2
EPYC	packet
Knights Mill	-