`specfem::attenuation::compute_integration_factors`¶

template<int N_SLS> IntegrationFactors<N_SLS> specfem::attenuation::compute_integration_factors(Kokkos::View<type_real[N_SLS], Kokkos::LayoutRight, Kokkos::HostSpace> tau_sigma, type_real deltat)¶

Compute Runge-Kutta coefficients for memory-variable time stepping.

Computes the fourth-order Runge-Kutta integration coefficients used to advance memory variables for viscoelastic attenuation. This function implements the computation of the integration coefficients (Savage et al. BSSA 2010, eq. 11).

For each SLS mechanism \( j \) with stress relaxation time \( \tau_{\sigma,j} \), the coefficients are computed as:

\[ \alpha_j = 1 - \frac{\Delta t}{\tau_{\sigma,j}} + \frac{(\Delta t)^2}{2\,\tau_{\sigma,j}^2} - \frac{(\Delta t)^3}{6\,\tau_{\sigma,j}^3} + \frac{(\Delta t)^4}{24\,\tau_{\sigma,j}^4} \]

\[ \beta_j = \frac{\Delta t}{2} - \frac{(\Delta t)^2}{3\,\tau_{\sigma,j}} + \frac{(\Delta t)^3}{8\,\tau_{\sigma,j}^2} - \frac{(\Delta t)^4}{24\,\tau_{\sigma,j}^3} \]

\[ \gamma_j = \frac{\Delta t}{2} - \frac{(\Delta t)^2}{6\,\tau_{\sigma,j}} + \frac{(\Delta t)^3}{24\,\tau_{\sigma,j}^2} \]

This notation is a departure from the Fortran implementation (and Savage et al. 2010), which sets the tauinv variable to \( - 1 / \tau_{\sigma,j} \) so that the sign switch is absorbed. Here, we follow a more standard mathematical convention where \( 1/\tau_{\sigma,j} \) remains positive, and the negative signs are explicitly included in the formula for the coefficients.

// Setup relaxation times and time step constexpr int N_SLS = 3;
Kokkos::View<type_real[N_SLS], Kokkos::LayoutRight, Kokkos::HostSpace>
tau_sigma("tau_sigma"); tau_sigma(0) = 1.0;  // 1 second tau_sigma(1) = 2.0;
// 2 seconds
tau_sigma(2) = 5.0;  // 5 seconds

type_real dt = 0.01; // 10 ms time step

// Compute integration factors auto factors =
specfem::attenuation::compute_integration_factors<N_SLS>( tau_sigma, dt);

// Use coefficients in time stepping for (int j = 0; j < N_SLS; ++j) { //
Apply Runge-Kutta update using factors.alpha(j), beta(j), gamma(j)
}

See also

tests/unit-tests/attenuation/compute_integration_factors_tests.cpp for detailed examples

Template Parameters:

N_SLS – Number of standard linear solids

Parameters:

tau_sigma – Stress relaxation times \( \tau_\sigma \) (positive values, seconds)
deltat – Time step \( \Delta t \) (seconds)

Returns:

IntegrationFactors containing the three coefficient arrays

template<int N_SLS> struct IntegrationFactors¶

Runge-Kutta memory-variable update coefficients per SLS mechanism.

Holds the three coefficient arrays \( \alpha_j \), \( \beta_j \), \( \gamma_j \) that advance each standard linear solid one time step via the low-storage fourth-order Runge-Kutta scheme of Savage et al. (BSSA 2010, eq. 11).

The coefficients are used to update memory variables in the attenuation implementation, providing stable time integration of the viscoelastic stress-strain relationships.

// Example usage with computed integration factors
constexpr int N_SLS = 3;
auto factors = compute_integration_factors<N_SLS>(tau_sigma, dt);

// Access coefficient arrays
for (int j = 0; j < N_SLS; ++j) {
  type_real alpha_j = factors.alpha(j);
  type_real beta_j  = factors.beta(j);
  type_real gamma_j = factors.gamma(j);
}

Template Parameters:: N_SLS – Number of standard linear solids

Public Members

Kokkos::View<type_real[N_SLS], Kokkos::LayoutRight, Kokkos::HostSpace> alpha¶: Fourth-order alpha coefficients.

Kokkos::View<type_real[N_SLS], Kokkos::LayoutRight, Kokkos::HostSpace> beta¶: Fourth-order beta coefficients.

Kokkos::View<type_real[N_SLS], Kokkos::LayoutRight, Kokkos::HostSpace> gamma¶: Fourth-order gamma coefficients.

specfem::attenuation::compute_integration_factors¶

`specfem::attenuation::compute_integration_factors`¶