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A New Class of Runge-Kutta Methods for Nonlinearly Partitioned Systems

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arxiv 2401.04859 v2 pith:GHQCLHQ7 submitted 2024-01-10 math.NA cs.NA

classification math.NAcs.NA
keywords methodsnprkrunge-kuttanonlinearlypartitionedclassintroducessolving
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This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned Runge-Kutta methods, and allow one to distribute different types of implicitness within nonlinear terms. The paper introduces the NPRK framework and discusses order conditions, linear stability, and the derivation of implicit-explicit and implicit-implicit NPRK integrators. The paper concludes with numerical experiments that demonstrate the utility of NPRK methods for solving viscous Burger's and the gray thermal radiation transport equations.

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  1. Preconditioning transformations of adjoint systems for evolution equations

    math.OC 2025-05 conditional novelty 7.0 of 10

    A symplectic framework for preconditioning adjoint systems preserves derivative backpropagation and enables stable scale-preconditioned optimization for coupled evolution equations.

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