Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, i... more Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, if they are not integrated in long-term earthquake cycle simulations. A widely applied procedure involves an 'overstressed asperity', a localized region stressed beyond the static frictional strength. The physical properties of the asperity (size, shape and overstress) may significantly impact rupture propagation. In particular, to induce a sustained rupture the asperity size needs to exceed a critical value. Although criteria for estimating the critical nucleation size under linear slip-weakening friction have been proposed for 2-D and 3-D problems based on simplifying assumptions, they do not provide general rules for designing 3-D numerical simulations. We conduct a parametric study to estimate parameters of the asperity that minimize numerical artefacts (e.g. changes of rupture shape and speed, artificial supershear transition, higher slip-rate amplitudes). We examine the critical size of square, circular and elliptical asperities as a function of asperity overstress and background (off-asperity) stress. For a given overstress, we find that asperity area controls rupture initiation while asperity shape is of lesser importance. The critical area obtained from our numerical results contrasts with published theoretical estimates when background stress is low. Therefore, we derive two new theoretical estimates of the critical size under low background stress while also accounting for overstress. Our numerical results suggest that setting the asperity overstress and area close to their critical values eliminates strong numerical artefacts even when the overstress is large. We also find that properly chosen asperity size or overstress may significantly shorten the duration of the initiation. Overall, our results provide guidelines for determining the size of the asperity and overstress to minimize the effects of the forced initiation on the subsequent spontaneous rupture propagation.
We have developed a 3D hybrid modeling technique based on the combination of the finite-element (... more We have developed a 3D hybrid modeling technique based on the combination of the finite-element (FE) and finite-difference (FD) methods to simulate dynamic rupture propagation on a fault, seismic wave radiation and propagation in a heterogeneous viscoelastic medium with realistic model of the attenuation. In an arbitrarily shaped part of the whole computational region, the 2nd-order displacement FE method is
The TSN (Traction-at-Split-Nodes) method has been developed independently by Andrews (1973, 1976,... more The TSN (Traction-at-Split-Nodes) method has been developed independently by Andrews (1973, 1976, 1999) and Day (1977, 1982). Andrews implemented his TSN formulation in the finite-difference scheme in which spatial differentiation is equivalent to the 2nd-order finite-element method. Day implemented his slightly different formulation of the TSN method in the 2nd-order partly-staggered finite-difference scheme. Dalguer and Day (2006) adapted the TSN method to the velocity-stress staggered-grid finite-difference scheme. Whereas the 4th-order spatial differencing is applied outside the fault, the 2nd-order differencing is applied along the fault plane. We present two implementations of the Day's TSN formulation in the velocity-stress staggered-grid finite-difference scheme for a 3D viscoelastic medium. In the first one we apply the 2nd-order spatial differencing everywhere in the grid including derivatives at the fault in the direction perpendicular to the fault plane. In the secon...
Geller and Takeuchi (1995) developed optimally accurate finite-difference (FD) operators. The ope... more Geller and Takeuchi (1995) developed optimally accurate finite-difference (FD) operators. The operators minimize the error of the numerical solution of the discretized equation of motion. The criterion for obtaining the optimally accurate operators requires that the leading term of the truncation error of the discretized homogeneous (without body-force term) equation of motion (that is if operand is an eigenfunction and frequency is equal to eigenfrequency) is zero. Consequently, the optimally accurate operators satisfy (up to the leading term of the truncation error) homogeneous equation of motion. The grid dispersion of an optimally accurate FD scheme is significantly smaller than that of a standard FD scheme. A heterogeneous FD scheme cannot be anything else than a FD approximation to the heterogeneous formulation of the equation of motion (the same form of the equation for a point away from a material discontinuity and a point at the material discontinuity). If an optimally accu...
Numerical prediction of earthquake ground motion in sedimentary basins and valleys often has to a... more Numerical prediction of earthquake ground motion in sedimentary basins and valleys often has to account for P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger, mainly in sediments below groundwater level. The ratio can attain values larger than 10 in unconsolidated sediments (e.g. in Ciudad de México). In a process of developing 3D optimally-accurate finite-difference schemes we encountered a serious problem with accuracy in media with large Vp/Vs ratio. This led us to investigate the very fundamental reasons for the inaccuracy. In order to identify the very basic inherent aspects of the numerical schemes responsible for their behavior with varying Vp/Vs ratio, we restricted to the most basic 2nd-order 2D numerical schemes on a uniform grid in a homogeneous medium. Although basic in the specified sense, the schemes comprise the decisive features for accuracy of wide class of numerical schemes. We investigated 6 numerical schemes: finite-difference_displacement_conve...
Simulated slip-rate time histories often are not free from spurious high-frequency oscillations. ... more Simulated slip-rate time histories often are not free from spurious high-frequency oscillations. This is because the used spatial grid is not fine enough to properly discretize possibly broad-spectrum slip-rate and stress variations and the spatial breakdown zone of the propagating rupture. In order to reduce the oscillations some numerical modelers apply the artificial damping. An alternative way is the application of the adaptive smoothing algorithm (ASA, Galis et al. 2010). The other modelers, however, rely on the a posteriori filtration. If the oscillations do not affect (change) development and propagation of the rupture during simulations, it is possible to apply a posteriori filtration to reduce the oscillations. Often, however, the a posteriori filtration is a problematic trade-off between suppression of oscillations and distortion of a true slip rate. We present quantitative comparison of efficiency of several methods. We have analyzed slip-rate time histories simulated by ...
An artificial stress asperity for initialization of spontaneous rupture propagation - a parametric study of a dynamic model with linear slip-weakening friction
Artificial procedures are used to initiate spontaneous rupture on faults with the linear slip-wea... more Artificial procedures are used to initiate spontaneous rupture on faults with the linear slip-weakening (LSW) friction law. Probably the most frequent technique is the stress asperity. It is important to minimize effects of the artificial initialization on the phase of the spontaneous rupture propagation. The effects may strongly depend on the geometry and size of the asperity, spatial distribution of the stress in and around the asperity, and a maximum stress-overshoot value. A square initialization zone with the stress discontinuously falling down at the asperity border to the level of the initial stress has been frequently applied (e.g., in the SCEC verification exercise). Galis et al. (2010) and Bizzarri (2010) independently introduced the elliptical asperity with a smooth spatial stress distribution in and around the asperity. In both papers the width of smoothing/tapering zone was only ad-hoc defined. Numerical simulations indicate that the ADER-DG method can account for a dis...
On Accuracy of the Finite-difference, Finite-element and Spectral-element Schemes for Modeling Seismic Motion in Media With a Large P-wave to S-wave Speed Ratio
The P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger often have to be accounte... more The P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger often have to be accounted for in numerical modeling of seismic motion in structurally and rheologically realistic models of sedimentary basins and valleys. Although sediments with large Vp/Vs usually do not make a major part of the computational region, their effect can be significant because they are at or very close to the free surface. However, the accuracy of the numerical schemes with respect to varying Vp/Vs is not often addressed in studies presenting schemes. In order to identify the very basic inherent aspects of the numerical schemes responsible for their behavior with varying Vp/Vs ratio, we included the most basic 2nd-order 2D numerical schemes on a uniform grid in a homogeneous medium. Although basic in the specified sense, the schemes comprise the decisive features for accuracy of wide class of numerical schemes. We also included 3D higher-order schemes. We investigated the following schemes (FD -...
Quantitative analysis of accuracy of seismic wave-propagation codes in 3D random scattering media
Several recent verification studies (e.g. Day et al., 2001; Bielak et al., 2010, Chaljub et al., ... more Several recent verification studies (e.g. Day et al., 2001; Bielak et al., 2010, Chaljub et al., 2010) have demonstrated the importance of assessing the accuracy of available numerical tools at low frequency in presence of large-scale features (basins, topography, etc.). The fast progress in high-performance computing, including efficient optimization of numerical codes on petascale supercomputers, has permitted the simulation of 3D seismic wave propagation at frequencies of engineering interest (up to 10Hz) in highly heterogeneous media (e.g. Hartzell et al, 2010; Imperatori and Mai, 2013). However, high frequency numerical simulations involving random scattering media, characterized by small-scale heterogeneities, are much more challenging for most numerical methods, and their verification may therefore be even more crucial than in the low-frequency case. Our goal is to quantitatively compare the accuracy and the behavior of three different numerical codes for seismic wave propaga...
Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, i... more Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, if they are not integrated in long-term earthquake cycle simulations. A widely applied procedure involves an 'overstressed asperity', a localized region stressed beyond the static frictional strength. The physical properties of the asperity (size, shape and overstress) may significantly impact rupture propagation. In particular, to induce a sustained rupture the asperity size needs to exceed a critical value. Although criteria for estimating the critical nucleation size under linear slip-weakening friction have been proposed for 2-D and 3-D problems based on simplifying assumptions, they do not provide general rules for designing 3-D numerical simulations. We conduct a parametric study to estimate parameters of the asperity that minimize numerical artefacts (e.g. changes of rupture shape and speed, artificial supershear transition, higher slip-rate amplitudes). We examine the critical size of square, circular and elliptical asperities as a function of asperity overstress and background (off-asperity) stress. For a given overstress, we find that asperity area controls rupture initiation while asperity shape is of lesser importance. The critical area obtained from our numerical results contrasts with published theoretical estimates when background stress is low. Therefore, we derive two new theoretical estimates of the critical size under low background stress while also accounting for overstress. Our numerical results suggest that setting the asperity overstress and area close to their critical values eliminates strong numerical artefacts even when the overstress is large. We also find that properly chosen asperity size or overstress may significantly shorten the duration of the initiation. Overall, our results provide guidelines for determining the size of the asperity and overstress to minimize the effects of the forced initiation on the subsequent spontaneous rupture propagation.
The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at ... more The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at boundaries of a grid in the finite-difference modeling of seismic wave propagation. We show relations between different formulations of the perfectly matched layer with respect to their three key aspects -split/unsplit, classical/convolutional, with the general/special form of the stretching factor. First we derive two variants of the split formulations for the general form of the stretching factor. Both variants naturally lead to the convolutional formulations in case of the general form of the stretching factor. One of them, L-split, reduces to the well-known classical split formulation in case of the special form of the stretching factor. The other, R-split, remains convolutional even for the special form of the stretching factor. The R-split formulation eventually leads to the equations identical with those obtained straightforwardly in the unsplit formulation.
The Southern California Earthquake Center (SCEC) organized the 3D Numerical Simulation Code Valid... more The Southern California Earthquake Center (SCEC) organized the 3D Numerical Simulation Code Validation Project for wave propagation in the past years. Recently, SCEC organizes an earthquake source physics code validation/comparison exercise. The goal of both efforts is to validate 3D earthquake simulation methods and foster their application by engineering community. Development of the earthquake motion numerical simulation methods is one of the primary goals of the Seismic Wave Propagation and Imaging in Complex Media: a European Network (SPICE, www.spice- rtn.org), the EU FP6 project. SPICE provides a reasonable platform for a code validation effort in Europe. We present here the SPICE Code Validation. The intention is to create a long-term basis for possible tests/comparisons/validation of numerical methods and codes for the earthquake motion simulation. The basis should serve even after the SPICE project is completed. Technically, the code validation process will be facilitated ...
We analyse 13 3D numerical time-domain explicit schemes for modelling seismic wave propagation an... more We analyse 13 3D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (V P /V S ). The 2 nd -order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The 4 th -order schemes include three finitedifference and two spectral-element schemes. All schemes are 2 nd -order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step, and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis.
We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave pr... more We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave propagation and earthquake motion in heterogeneous media. The method is based on a combination of the fourth-order velocity-stress staggered-grid finite-difference (FD) scheme, that covers a major part of a computational domain, with the second-order finite-element (FE) method which can be applied to one or several relatively small subdomains. The FD and FE parts causally communicate at each time level in the FD-FE transition zone consisting of the FE Dirichlet boundary, FD-FE averaging zone and FD Dirichlet zone.
Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, i... more Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, if they are not integrated in long-term earthquake cycle simulations. A widely applied procedure involves an 'overstressed asperity', a localized region stressed beyond the static frictional strength. The physical properties of the asperity (size, shape and overstress) may significantly impact rupture propagation. In particular, to induce a sustained rupture the asperity size needs to exceed a critical value. Although criteria for estimating the critical nucleation size under linear slip-weakening friction have been proposed for 2-D and 3-D problems based on simplifying assumptions, they do not provide general rules for designing 3-D numerical simulations. We conduct a parametric study to estimate parameters of the asperity that minimize numerical artefacts (e.g. changes of rupture shape and speed, artificial supershear transition, higher slip-rate amplitudes). We examine the critical size of square, circular and elliptical asperities as a function of asperity overstress and background (off-asperity) stress. For a given overstress, we find that asperity area controls rupture initiation while asperity shape is of lesser importance. The critical area obtained from our numerical results contrasts with published theoretical estimates when background stress is low. Therefore, we derive two new theoretical estimates of the critical size under low background stress while also accounting for overstress. Our numerical results suggest that setting the asperity overstress and area close to their critical values eliminates strong numerical artefacts even when the overstress is large. We also find that properly chosen asperity size or overstress may significantly shorten the duration of the initiation. Overall, our results provide guidelines for determining the size of the asperity and overstress to minimize the effects of the forced initiation on the subsequent spontaneous rupture propagation.
We have developed a 3D hybrid modeling technique based on the combination of the finite-element (... more We have developed a 3D hybrid modeling technique based on the combination of the finite-element (FE) and finite-difference (FD) methods to simulate dynamic rupture propagation on a fault, seismic wave radiation and propagation in a heterogeneous viscoelastic medium with realistic model of the attenuation. In an arbitrarily shaped part of the whole computational region, the 2nd-order displacement FE method is
The TSN (Traction-at-Split-Nodes) method has been developed independently by Andrews (1973, 1976,... more The TSN (Traction-at-Split-Nodes) method has been developed independently by Andrews (1973, 1976, 1999) and Day (1977, 1982). Andrews implemented his TSN formulation in the finite-difference scheme in which spatial differentiation is equivalent to the 2nd-order finite-element method. Day implemented his slightly different formulation of the TSN method in the 2nd-order partly-staggered finite-difference scheme. Dalguer and Day (2006) adapted the TSN method to the velocity-stress staggered-grid finite-difference scheme. Whereas the 4th-order spatial differencing is applied outside the fault, the 2nd-order differencing is applied along the fault plane. We present two implementations of the Day's TSN formulation in the velocity-stress staggered-grid finite-difference scheme for a 3D viscoelastic medium. In the first one we apply the 2nd-order spatial differencing everywhere in the grid including derivatives at the fault in the direction perpendicular to the fault plane. In the secon...
Geller and Takeuchi (1995) developed optimally accurate finite-difference (FD) operators. The ope... more Geller and Takeuchi (1995) developed optimally accurate finite-difference (FD) operators. The operators minimize the error of the numerical solution of the discretized equation of motion. The criterion for obtaining the optimally accurate operators requires that the leading term of the truncation error of the discretized homogeneous (without body-force term) equation of motion (that is if operand is an eigenfunction and frequency is equal to eigenfrequency) is zero. Consequently, the optimally accurate operators satisfy (up to the leading term of the truncation error) homogeneous equation of motion. The grid dispersion of an optimally accurate FD scheme is significantly smaller than that of a standard FD scheme. A heterogeneous FD scheme cannot be anything else than a FD approximation to the heterogeneous formulation of the equation of motion (the same form of the equation for a point away from a material discontinuity and a point at the material discontinuity). If an optimally accu...
Numerical prediction of earthquake ground motion in sedimentary basins and valleys often has to a... more Numerical prediction of earthquake ground motion in sedimentary basins and valleys often has to account for P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger, mainly in sediments below groundwater level. The ratio can attain values larger than 10 in unconsolidated sediments (e.g. in Ciudad de México). In a process of developing 3D optimally-accurate finite-difference schemes we encountered a serious problem with accuracy in media with large Vp/Vs ratio. This led us to investigate the very fundamental reasons for the inaccuracy. In order to identify the very basic inherent aspects of the numerical schemes responsible for their behavior with varying Vp/Vs ratio, we restricted to the most basic 2nd-order 2D numerical schemes on a uniform grid in a homogeneous medium. Although basic in the specified sense, the schemes comprise the decisive features for accuracy of wide class of numerical schemes. We investigated 6 numerical schemes: finite-difference_displacement_conve...
Simulated slip-rate time histories often are not free from spurious high-frequency oscillations. ... more Simulated slip-rate time histories often are not free from spurious high-frequency oscillations. This is because the used spatial grid is not fine enough to properly discretize possibly broad-spectrum slip-rate and stress variations and the spatial breakdown zone of the propagating rupture. In order to reduce the oscillations some numerical modelers apply the artificial damping. An alternative way is the application of the adaptive smoothing algorithm (ASA, Galis et al. 2010). The other modelers, however, rely on the a posteriori filtration. If the oscillations do not affect (change) development and propagation of the rupture during simulations, it is possible to apply a posteriori filtration to reduce the oscillations. Often, however, the a posteriori filtration is a problematic trade-off between suppression of oscillations and distortion of a true slip rate. We present quantitative comparison of efficiency of several methods. We have analyzed slip-rate time histories simulated by ...
An artificial stress asperity for initialization of spontaneous rupture propagation - a parametric study of a dynamic model with linear slip-weakening friction
Artificial procedures are used to initiate spontaneous rupture on faults with the linear slip-wea... more Artificial procedures are used to initiate spontaneous rupture on faults with the linear slip-weakening (LSW) friction law. Probably the most frequent technique is the stress asperity. It is important to minimize effects of the artificial initialization on the phase of the spontaneous rupture propagation. The effects may strongly depend on the geometry and size of the asperity, spatial distribution of the stress in and around the asperity, and a maximum stress-overshoot value. A square initialization zone with the stress discontinuously falling down at the asperity border to the level of the initial stress has been frequently applied (e.g., in the SCEC verification exercise). Galis et al. (2010) and Bizzarri (2010) independently introduced the elliptical asperity with a smooth spatial stress distribution in and around the asperity. In both papers the width of smoothing/tapering zone was only ad-hoc defined. Numerical simulations indicate that the ADER-DG method can account for a dis...
On Accuracy of the Finite-difference, Finite-element and Spectral-element Schemes for Modeling Seismic Motion in Media With a Large P-wave to S-wave Speed Ratio
The P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger often have to be accounte... more The P-wave to S-wave speed ratios (Vp/Vs) as large as 5 and even larger often have to be accounted for in numerical modeling of seismic motion in structurally and rheologically realistic models of sedimentary basins and valleys. Although sediments with large Vp/Vs usually do not make a major part of the computational region, their effect can be significant because they are at or very close to the free surface. However, the accuracy of the numerical schemes with respect to varying Vp/Vs is not often addressed in studies presenting schemes. In order to identify the very basic inherent aspects of the numerical schemes responsible for their behavior with varying Vp/Vs ratio, we included the most basic 2nd-order 2D numerical schemes on a uniform grid in a homogeneous medium. Although basic in the specified sense, the schemes comprise the decisive features for accuracy of wide class of numerical schemes. We also included 3D higher-order schemes. We investigated the following schemes (FD -...
Quantitative analysis of accuracy of seismic wave-propagation codes in 3D random scattering media
Several recent verification studies (e.g. Day et al., 2001; Bielak et al., 2010, Chaljub et al., ... more Several recent verification studies (e.g. Day et al., 2001; Bielak et al., 2010, Chaljub et al., 2010) have demonstrated the importance of assessing the accuracy of available numerical tools at low frequency in presence of large-scale features (basins, topography, etc.). The fast progress in high-performance computing, including efficient optimization of numerical codes on petascale supercomputers, has permitted the simulation of 3D seismic wave propagation at frequencies of engineering interest (up to 10Hz) in highly heterogeneous media (e.g. Hartzell et al, 2010; Imperatori and Mai, 2013). However, high frequency numerical simulations involving random scattering media, characterized by small-scale heterogeneities, are much more challenging for most numerical methods, and their verification may therefore be even more crucial than in the low-frequency case. Our goal is to quantitatively compare the accuracy and the behavior of three different numerical codes for seismic wave propaga...
Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, i... more Numerical simulations of dynamic earthquake rupture require an artificial initiation procedure, if they are not integrated in long-term earthquake cycle simulations. A widely applied procedure involves an 'overstressed asperity', a localized region stressed beyond the static frictional strength. The physical properties of the asperity (size, shape and overstress) may significantly impact rupture propagation. In particular, to induce a sustained rupture the asperity size needs to exceed a critical value. Although criteria for estimating the critical nucleation size under linear slip-weakening friction have been proposed for 2-D and 3-D problems based on simplifying assumptions, they do not provide general rules for designing 3-D numerical simulations. We conduct a parametric study to estimate parameters of the asperity that minimize numerical artefacts (e.g. changes of rupture shape and speed, artificial supershear transition, higher slip-rate amplitudes). We examine the critical size of square, circular and elliptical asperities as a function of asperity overstress and background (off-asperity) stress. For a given overstress, we find that asperity area controls rupture initiation while asperity shape is of lesser importance. The critical area obtained from our numerical results contrasts with published theoretical estimates when background stress is low. Therefore, we derive two new theoretical estimates of the critical size under low background stress while also accounting for overstress. Our numerical results suggest that setting the asperity overstress and area close to their critical values eliminates strong numerical artefacts even when the overstress is large. We also find that properly chosen asperity size or overstress may significantly shorten the duration of the initiation. Overall, our results provide guidelines for determining the size of the asperity and overstress to minimize the effects of the forced initiation on the subsequent spontaneous rupture propagation.
The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at ... more The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at boundaries of a grid in the finite-difference modeling of seismic wave propagation. We show relations between different formulations of the perfectly matched layer with respect to their three key aspects -split/unsplit, classical/convolutional, with the general/special form of the stretching factor. First we derive two variants of the split formulations for the general form of the stretching factor. Both variants naturally lead to the convolutional formulations in case of the general form of the stretching factor. One of them, L-split, reduces to the well-known classical split formulation in case of the special form of the stretching factor. The other, R-split, remains convolutional even for the special form of the stretching factor. The R-split formulation eventually leads to the equations identical with those obtained straightforwardly in the unsplit formulation.
The Southern California Earthquake Center (SCEC) organized the 3D Numerical Simulation Code Valid... more The Southern California Earthquake Center (SCEC) organized the 3D Numerical Simulation Code Validation Project for wave propagation in the past years. Recently, SCEC organizes an earthquake source physics code validation/comparison exercise. The goal of both efforts is to validate 3D earthquake simulation methods and foster their application by engineering community. Development of the earthquake motion numerical simulation methods is one of the primary goals of the Seismic Wave Propagation and Imaging in Complex Media: a European Network (SPICE, www.spice- rtn.org), the EU FP6 project. SPICE provides a reasonable platform for a code validation effort in Europe. We present here the SPICE Code Validation. The intention is to create a long-term basis for possible tests/comparisons/validation of numerical methods and codes for the earthquake motion simulation. The basis should serve even after the SPICE project is completed. Technically, the code validation process will be facilitated ...
We analyse 13 3D numerical time-domain explicit schemes for modelling seismic wave propagation an... more We analyse 13 3D numerical time-domain explicit schemes for modelling seismic wave propagation and earthquake motion for their behaviour with a varying P-wave to S-wave speed ratio (V P /V S ). The 2 nd -order schemes include three finite-difference, three finite-element and one discontinuous-Galerkin schemes. The 4 th -order schemes include three finitedifference and two spectral-element schemes. All schemes are 2 nd -order in time. We assume a uniform cubic grid/mesh and present all schemes in a unified form. We assume plane S wave propagation in an unbounded homogeneous isotropic elastic medium. We define relative local errors of the schemes in amplitude and the vector difference in one time step, and normalize them for a unit time. We also define the equivalent spatial sampling ratio as a ratio at which the maximum relative error is equal to the reference maximum error. We present results of the extensive numerical analysis.
We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave pr... more We have developed a new hybrid numerical method for 3-D viscoelastic modelling of seismic wave propagation and earthquake motion in heterogeneous media. The method is based on a combination of the fourth-order velocity-stress staggered-grid finite-difference (FD) scheme, that covers a major part of a computational domain, with the second-order finite-element (FE) method which can be applied to one or several relatively small subdomains. The FD and FE parts causally communicate at each time level in the FD-FE transition zone consisting of the FE Dirichlet boundary, FD-FE averaging zone and FD Dirichlet zone.
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Papers by Martin Galis