An energy-based discontinuous Galerkin discretization of the elastic wave equation in second order form

2018 ◽  
Vol 338 ◽  
pp. 362-391 ◽  
Author(s):  
Daniel Appelö ◽  
Thomas Hagstrom
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Discontinuous Galerkin ◽  
Second Order ◽  
Elastic Wave Equation ◽  
Galerkin Discretization ◽  
Order Form

scholarly journals Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

2019 ◽  
Vol 219 (3) ◽  
pp. 1900-1914 ◽  
Author(s):  
T Möller ◽  
W Friederich
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Discontinuous Galerkin ◽  
Heterogeneous Media ◽  
Rheological Behaviour ◽  
Displacement Discontinuity ◽  
Signal Frequency ◽  
Elastic Wave Equation ◽  
Numerical Flux

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

2007 ◽  
Vol 45 (5) ◽  
pp. 1902-1936 ◽  
Author(s):  
Stefan Nilsson ◽  
N. Anders Petersson ◽  
Björn Sjögreen ◽  
Heinz-Otto Kreiss
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Second Order ◽  
Elastic Wave Equation ◽  
Difference Approximations
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