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Extreme-scale Discontinuous Galerkin Environment (EDGE)

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# EDGƎ

## Equations

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
``````

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
``````

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.

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
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

Extreme-scale Discontinuous Galerkin Environment (EDGE)

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