2.4.11.4. MRIStep Custom Inner Steppers

Recall that infinitesimal multirate methods require solving a set of auxiliary IVPs

(2.76)\[\dot{v}(t) = f^F(t, v) + r_i(t), \qquad v(t_{i,0}) = v_{i,0},\]

on intervals \(t \in [t_{i,0}, t_{i,f}]\). For the MIS, MRI-GARK and IMEX-MRI-GARK methods implemented in MRIStep, the forcing term \(r_i(t)\) presented in §2.2.7 can be equivalently written as

(2.77)\[r_i(t) = \sum\limits_{k \geq 1} \hat{\omega}_{i,k} \tau^{k-1} + \sum\limits_{k \geq 1} \hat{\gamma}_{i,k} \tau^{k-1}\]

where \(\tau = (t - t_{n,i-1}^S)/(h^S \Delta c_i^S)\) is the normalized time with \(\Delta c_i^S=\left(c^S_i - c^S_{i-1}\right)\), the slow stage times are \(t_{n,i-1}^S = t_{n-1} + c_{i-1}^S h^S\), and the polynomial coefficient vectors are

(2.78)\[\hat{\omega}_{i,k} = \frac{1}{\Delta c_i^S} \sum\limits_{j=1}^{i-1} \Omega_{i,j,k} f^E(t_{n,j}^S, z_j) \quad\text{and}\quad \hat{\gamma}_{i,k} = \frac{1}{\Delta c_i^S} \sum\limits_{j=1}^i \Gamma_{i,j,k} f^I(t_{n,j}^S, z_j).\]

The MERK and IMEX-MRI-SR methods included in MRIStep compute the forcing polynomial (2.77) similarly, with appropriate modifications to \(\Delta c_i^S\), \(t_{n,i-1}^S\), and the coefficients (2.78).

To evolve the IVP (2.76) MRIStep utilizes a generic time integrator interface defined by the MRIStepInnerStepper base class. This section presents the MRIStepInnerStepper base class and methods that define the integrator interface as well as detailing the steps for creating an MRIStepInnerStepper.