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1. DIRK Schemes with High Weak Stage Order

   https://doi.org/10.1007/978-3-030-39647-3_36  

   Ketcheson, D.; Seibold, B.; Shirokoff, D.; Zhou, D. (August 2020, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018. Lecture Notes in Computational Science and Engineering. Springer.)

   Sherwin, S.; Moxey, D.; Peiro, J.; Vincent, P.; Schwab, C. (Ed.)

   Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples. 

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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. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

   https://doi.org/10.1007/s42967-021-00179-6  

   Li, Liang; Zhu, Jun; Shu, Chi-Wang; Zhang, Yong-Tao (March 2022, Communications on Applied Mathematics and Computation)

   Abstract Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions. 

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4. 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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5. 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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