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1. Spatial manifestations of order reduction in Runge–Kutta methods for initial boundary value problems

   https://doi.org/10.4310/CMS.2024.v22.n3.a2  

   Rosales, Rodolfo Ruben; Seibold, Benjamin; Shirokoff, David; Zhou, Dong (January 2024, Communications in Mathematical Sciences)

   This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge–Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed. 

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2. Explicit Runge–Kutta Methods that Alleviate Order Reduction

   https://doi.org/10.1137/23M1606812  

   Biswas, Abhijit; Ketcheson, David I; Roberts, Steven; Seibold, Benjamin; Shirokoff, David (August 2025, SIAM Journal on Numerical Analysis)

   Explicit Runge--Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to an initial boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method's order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems. 

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3. Irksome: Automating Runge–Kutta Time-stepping for Finite Element Methods

   https://doi.org/10.1145/3466168  

   Farrell, Patrick E.; Kirby, Robert C.; Marchena-Menéndez, Jorge (December 2021, ACM Transactions on Mathematical Software)

   While implicit Runge–Kutta (RK) methods possess high order accuracy and important stability properties, implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as impediments. We present Irksome , a high-level library for manipulating UFL (Unified Form Language) expressions of semidiscrete variational forms to obtain UFL expressions for the coupled Runge–Kutta stage equations at each time step. Irksome works with the Firedrake package to enable the efficient solution of the resulting coupled algebraic systems. Numerical examples confirm the efficacy of the software and our solver techniques for various problems. 

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4. Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

   https://doi.org/10.24297/jam.v17i0.8408  

   Chowdhury, Abhinandan; Clayton, Sammie; Lemma, Mulatu (July 2019, JOURNAL OF ADVANCES IN MATHEMATICS)

   We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered. 

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5. Invariant-Domain Preserving High-Order Time Stepping: II. IMEX Schemes

   https://doi.org/10.1137/22M1505025  

   Ern, Alexandre; Guermond, Jean-Luc (October 2023, SIAM Journal on Scientific Computing)

   We consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. We propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariant-domain preserving and mass conservative. Following the ideas introduced in Part I on explicit Runge--Kutta schemes, the IMEX scheme is written in incremental form. At each stage, we first combine a low-order and a high-order hyperbolic update using a limiting operator, then we combine a low-order and a high-order parabolic update using another limiting operator. The proposed technique, which is agnostic to the space discretization, allows one to optimize the time step restrictions induced by the hyperbolic substep. To illustrate the proposed methodology, we derive four novel IMEX methods with optimal efficiency. All the implicit schemes are singly diagonal. One of them is A-stable and the other three are L-stable. The novel IMEX schemes are evaluated numerically on systems of stiff ordinary differential equations and nonlinear conservation equations. 

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