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1. Structure aware Runge–Kutta time stepping for spacetime tents

   https://doi.org/10.1007/s42985-020-00020-4  

   Gopalakrishnan, Jay; Schöberl, Joachim; Wintersteiger, Christoph (August 2020, SN Partial Differential Equations and Applications)

   null (Ed.)

   Abstract We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations. 

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2. 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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3. 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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4. IMEX Runge-Kutta Parareal for Non-diffusive Equations

   https://doi.org/10.1007/978-3-030-75933-9_5  

   Buvoli, Tommaso; Minion, Michael (August 2021, Parallel-in-Time Integration Methods)

   Ong, B.; Schroder, J.; Shipton, J.; Friedhoff, S (Ed.)

   Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations possess low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrödinger equation demonstrate the analytical conclusions. 

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5. A High-Order Eulerian–Lagrangian Runge–Kutta Finite Volume (EL–RK–FV) Method for Scalar Nonlinear Conservation Laws

   https://doi.org/10.1007/s10915-024-02714-y  

   Chen, Jiajie; Nakao, Joseph; Qiu, Jing-Mei; Yang, Yang (November 2024, Journal of Scientific Computing)

   Abstract We present a class of high-order Eulerian–Lagrangian Runge–Kutta finite volume methods that can numerically solve Burgers’ equation with shock formations, which could be extended to general scalar conservation laws. Eulerian–Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine–Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge–Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme’s high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes. 

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