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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
  

• elastic (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
  

• elastic (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
  

• viscoelastic (2D)

  Solves the two-dimensional elastic wave equations with frequency-independent attenuation. The vector of quantities q(x,y,t)=(sigma_xx, sigma_yy, sigma_xy, u, v, m_11, m_12, m_13, ..., m_n1, m_n2, m_n3) contains the elastic quantities and additional memory variables m_11, ..., m_n3. n gives the number of relaxation mechanisms with three quantities per mechanism. The Jacobians A(x,y) and B(x,y) are allowed to be set per element and summarize the material parameters. The matrix E(x,y) is the reactive source term.

  q_t + A  q_x + B q_y = E
  

• viscoelastic (3D)

  Solves the three-dimensional elastic wave equations with frequency-independent attenuation. The vector of quantities q(x,y,z,t)=(sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_xz, sigma_yz, u, v, w, m_11, ..., m_16, ..., m_n1, ..., m_n6) contains the elastic quantities and additional memory variables m_11, ..., m_n6. n gives the number of relaxation mechanisms with six quantities per mechanism. 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. The matrix E(x,y,z) is the reactive source term.

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

• 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 tetrahedralizations 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
elastic	quad4r, tria3, hex8r, tet4	x	x	x	x
viscoelastic	quad4r, tria3, hex8r, tet4	x	x	x	x
swe	line, quad4r, tria3	x	x

High Performance Support

Microarchitecture	Machine(s)
Haswell	Comet
Knights Landing	Stampede 2, Cori Phase 2, Theta
Knights Mill	-
Skylake	Amazon Elastic Compute Cloud, Google Cloud Platform, Stampede 2
Cascade Lake	Frontera
Ice Lake	Oracle Cloud Infrastructure
Rome	Oracle Cloud Infrastructure
Milan	Oracle Cloud Infrastructure
Neoverse N1	Amazon Elastic Compute Cloud, Oracle Cloud Infrastructure