This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this flexibility at the slow time scale  , the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified ``fast'' initial-value problems, that may be solved using any viable solver . We name the proposed class as implicit-explicit multirate infinitesimal (IMEX-MRI ) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of generalized-structure additive Runge--Kutta (GARK) methods, and derive a set of order conditions on the IMEX-MRI coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI methods, two of order three and one of order four. We conclude with numerical simulations demonstrating their predicted convergence rates on two multirate test problems, and compare their efficiency against  legacy IMEX multirate  methods.
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility , the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm . We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK ) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of generalized-structure additive Runge--Kutta (GARK) methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.
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flexibility at the
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flexibility
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``fast''
`fast'
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solver
algorithm
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(IMEX-MRI
generalized-structure additive Runge--Kutta (IMEX-MRI-GARK
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IMEX-MRI
IMEX-MRI-GARK
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IMEX-MRI
IMEX-MRI-GARK
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demonstrating their predicted convergence rates
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and compare
demonstrating the methods' predicted convergence rates and comparing
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both
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schemes and recent third and fourth order implicit MRI-GARK