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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. Third-Order Sectorially A-Stable Alternating Implicit Runge–Kutta Schemes

   https://doi.org/10.1137/24M1650624  

   Ern, Alexandre; Guermond, Jean-Luc (June 2025, SIAM Journal on Scientific Computing)

   We design pairs of six-stage, third-order, alternating implicit Runge–Kutta (RK) schemes that can be used to integrate in time two stiff operators by an operator-splitting technique. We also design for each pair a companion explicit RK scheme to be used for a third, nonstiff oper- ator in an implicit-explicit (IMEX) fashion. The main application we have in mind is (non)linear parabolic problems, where the two stiff operators represent diffusion processes (for instance, in two spatial directions) and the nonstiff operator represents (non)linear transport. We identify necessary conditions for linear sectorial A( )-stability by considering a scalar ODE with two (complex) ei- genvalues lying in some fixed cone of the half-complex plane with nonpositive real part. We show numerically that it is possible to achieve A(0)-stability when combining two operators with negative eigenvalues, irrespective of their relative magnitude. Finally, we show by numerical examples includ- ing two-dimensional nonlinear transport problems discretized in space using finite elements that the proposed schemes behave well. 

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3. Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

   https://doi.org/10.1051/m2an/2018069  

   Zhou, Lingling; Xia, Yinhua; Shu, Chi-Wang (January 2019, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh 2 and the second order TVD-RK scheme needs $$ \tau \le \rho {h}^{\frac{4}{3}}$$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h . 

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4. 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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5. Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

   https://doi.org/10.1093/mnras/stac1257  

   Zhang, Weijie; Xing, Yulong; Endeve, Eirik (May 2022, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation. 

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