Related papers: Symplectic Isospectral Runge--Kutta Methods as Lie…
We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density…
Exponential Runge--Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge--Kutta methods, however, relies on a convergence…
We present MDIRK: a Multifluid second-order Diagonally-Implicit Runge-Kutta method to study momentum transfer between gas and an arbitrary number ($N$) of dust species. The method integrates the equations of hydrodynamics with an Implicit…
The residual-based variational multiscale (VMS) formulation has achieved remarkable success in large-eddy simulation of turbulent flows. However, its temporal discretization has largely remained limited to second-order implicit schemes. The…
We show that the isomorphism between the moduli space of certain parabolic Higgs bundles over an elliptic curve and the Hilbert scheme of n points of the cotangent bundle of the elliptic curve is a symplectomorphism with respect to their…
In this letter, we revisit the IEQ method and provide a new perspective on its ability to preserve the original energy dissipation laws. The invariant energy quadratization (IEQ) method has been widely used to design energy stable numerical…
Compact Runge-Kutta (cRK) Flux Reconstruction (FR) methods are a variant of RKFR methods for hyperbolic conservation laws with a compact stencil including only immediate neighboring finite elements. We extend cRKFR methods to handle…
A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called…
Symmetry is a key property of numerical methods. The geometric properties of symmetric schemes make them an attractive option for integrating Hamiltonian systems, whilst their ability to exactly recover the initial condition without the…
We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the…
Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on…
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic Gamma distributed DDEs are not currently available. Accordingly, modellers often resort…
The family of PDE-constrained LDDMM methods is emerging as a particularly interesting approach for physically meaningful diffeomorphic transformations. The original combination of Gauss--Newton--Krylov optimization and Runge--Kutta…
It is well known that the integrable Hamiltonian systems defined by the Adler-Kostant-Symes construction correspond via Hamiltonian reduction to systems on cotangent bundles of Lie groups. Generalizing previous results on Toda systems, here…
We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie…
We study spatially partitioned embedded Runge--Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient…
This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank…
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the…
A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic…
In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a non-autonomous ordinarily differential equation (ODE) found in improving convergence rate of Nesterov's accelerated gradient method.…