Related papers: Probabilistic ODE Solvers with Runge-Kutta Means
This paper deals with stability of classical Runge-Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [12] and used in [13] for the time integration of nonlinear evolution PDEs, the…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
A new format for commutator-free Lie group methods is proposed based on explicit classical Runge-Kutta schemes. In this format exponentials are reused at every stage and the storage is required only for two quantities: the right hand side…
Numerical ordinary differential equation (ODE) solvers are indispensable tools in various engineering domains, enabling the simulation and analysis of dynamic systems. In this work, we utilize 5 different numerical ODE solvers namely:…
In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent…
Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit…
The class of stochastic Runge-Kutta methods for stochastic differential equations due to R\"o{\ss}ler is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are…
We propose a new method to solve the relativistic hydrodynamic equations based on implicit Runge-Kutta methods with a locally optimized fixed-point iterative solver. For numerical demonstration, we implement our idea for ideal hydrodynamics…
Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study…
In the present paper, we introduce a new family of $ \theta-$methods for solving delay differential equations. New methods are developed using a combination of decomposition technique viz. new iterative method proposed by Daftardar Gejji…
A general class of stochastic Runge-Kutta methods for the weak approximation of It\^o and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an…
Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and…
Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially…
In this note, we connect two different topics from linear algebra and numerical analysis: hypocoercivity of semi-dissipative matrices and strong stability for explicit Runge--Kutta schemes. Linear autonomous ODE systems with a non-coercive…
We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK)…
The properties of the Bigeometric or proportional derivative are presented and discussed explicitly. Based on this derivative, the Bigeometric Taylor theorem is worked out. As an application of this calculus, the Bigeometric Runge-Kutta…
A focus of recent research in quantum computing has been on developing quantum algorithms for differential equations solving using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which…
This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit-explicit Runge-Kutta (IMEX-RK) discretization of the semi-discrete equations.…
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the…
We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge--Kutta pairs. We consider orders four through twelve, including both serial and parallel…