Related papers: High order linearly implicit methods for evolution…
Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
Rational methods are intended to time integrate linear homogeneous problems. However, their scope can be extended so as to cover linear nonhomogeneous problems. In this paper the integration of semilinear problems is considered. The…
This paper considers the approximation of partial differential equations with a point collocation framework based on high-order local maximum-entropy schemes (HOLMES). In this approach, smooth basis functions are computed through an…
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…
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…
This chapter provides an introduction to Hybrid High-Order (HHO) methods. These are new generation numerical methods for PDEs with several advantageous features: the support of arbitrary approximation orders on general polyhedral meshes,…
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time…
Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this paper, we derive explicit stabilized integrators of orders one and…
Review of implicit methods of integrating system of stiff ordinary differential equations is presented. Defines and graphically presents absolute stability region for Gears methods (backward differentiation formula) used to solve system of…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems,…
It is well known that multigrid methods are optimally efficient for solution of elliptic equations (O(N)), which means that effort is proportional to the number of points at which the solution is evaluated). Thus this is an ideal method to…
The paper studies several approaches to numerical integration over a domain defined implicitly by an indicator function such as the level set function. The integration methods are based on subdivision, moment--fitting, local…
This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions…
A combination of implicit and explicit timestepping is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. Loosely speaking, the method we consider is implicit…
Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations…
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…
In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential…
This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…