More Like this

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. 

   more » « less
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. 

   more » « less
3. The Physics of Pair-Density Waves: Cuprate Superconductors and Beyond

   https://doi.org/10.1146/annurev-conmatphys-031119-050711  

   Agterberg, Daniel F.; Davis, J.C. Séamus; Edkins, Stephen D.; Fradkin, Eduardo; Van Harlingen, Dale J.; Kivelson, Steven A.; Lee, Patrick A.; Radzihovsky, Leo; Tranquada, John M.; Wang, Yuxuan (March 2020, Annual Review of Condensed Matter Physics)

   We review the physics of pair-density wave (PDW) superconductors. We begin with a macroscopic description that emphasizes order induced by PDW states, such as charge-density wave, and discuss related vestigial states that emerge as a consequence of partial melting of the PDW order. We review and critically discuss the mounting experimental evidence for such PDW order in the cuprate superconductors, the status of the theoretical microscopic description of such order, and the current debate on whether the PDW is a mother order or another competing order in the cuprates. In addition, we give an overview of the weak coupling version of PDW order, Fulde–Ferrell–Larkin–Ovchinnikov states, in the context of cold atom systems, unconventional superconductors, and noncentrosymmetric and Weyl materials. 

   more » « less
4. Ray class groups and ray class fields for orders of number fields

   https://doi.org/10.2140/ent.2025.4.1  

   Kopp, Gene S; Lagarias, Jeffrey C (February 2025, Essential Number Theory)

   We contribute to the theory of orders of number fields. We define a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. We show existence of ray class fields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. We give exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes. 

   more » « less
5. Finite element-based invariant-domain preserving approximation of hyperbolic systems: Beyond second-order accuracy in space

   https://doi.org/10.1016/j.cma.2023.116470  

   Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan (January 2024, Computer Methods in Applied Mechanics and Engineering)

   This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. 

   more » « less