Related papers: High-Order, Implicit Time Integration of Discrete,…
The precise description of quantum nuclear fluctuations in atomistic modelling is possible by employing path integral techniques, which involve a considerable computational overhead due to the need of simulating multiple replicas of the…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators…
A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In…
Numerical schemes used for the integration of complex flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, the performance of various high-order accurate numerical schemes is…
The single-step explicit time integration methods have long been valuable for solving large-scale nonlinear structural dynamic problems, classified into single-solve and multi-sub-step approaches. However, no existing explicit single-solve…
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…
High-order adaptive time-stepping algorithms are of significant practical value and theoretical interest for accelerating long-time fluid-flow simulations and resolving complex dynamical behaviors. While several high-order implicit-explicit…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
Many HPC applications that solve differential equations rely on the Runge-Kutta family of methods for time integration. Among these methods, the fourth-order accurate RK4 scheme is especially popular. This time integration scheme requires…
Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis…
Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models…
We show in this paper that third- and fourth-order low storage Runge-Kutta algorithms can be built specifically for quadratic nonlinear operators, at the expense of roughly doubling the time needed for evaluating the temporal derivatives.…
By a high-order numerical homogenization method, a heterogeneous multiscale scheme was developed in Jin & Li (2022) for evolving differential equations containing two time scales. In this paper, we further explore the technique to propose…
Projection-based nonlinear model order reduction methods can be used to reduce simulation times for the solution of many PDE-constrained problems. It has been observed in literature that such nonlinear reduced-order models (ROMs) based on…
In this work, we present a new high order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the…
Most numerical methods for time integration use real time steps. Complex time steps provide an additional degree of freedom, as we can select the magnitude of the step in both the real and imaginary directions. By time stepping along…
In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this…
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory and astrophysical plasmas. To solve these equations, massively parallel codes have been developed and run on present-day supercomputers. This paper describes…
This paper proposes a strong second-order two-step explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for solving a two-dimensional Gray-Scott model subject to appropriate initial and boundary…
First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored…