Related papers: Exponential integrators for second-order in time p…
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.
In this paper, we report about recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs), by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In…
The implementation of the discrete adjoint method for exponential time differencing (ETD) schemes is considered. This is important for parameter estimation problems that are constrained by stiff time-dependent PDEs when the discretized PDE…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
In this paper we apply the boundary elements method (BEM) and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of two-dimensional time-fractional partial differential equations (TFPDEs). The fractional…
Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular…
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems,…
Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…
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…
The goal of this project is to compare the performance of exponential time integrators with traditional methods such as diagonally implicit Runge-Kutta methods in the context of solving the system of reduced magnetohydrodynamics (RMHD). In…
Time integration of advection dominated advection-diffusion problems on refined meshes can be a challenging task, since local refinement can lead to a severe time step restriction, whereas standard implicit time stepping is usually hardly…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of…
This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit…
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial…
This paper presents a novel approach to rigorously solving initial value problems for semilinear parabolic partial differential equations (PDEs) using fully spectral Fourier-Chebyshev expansions. By reformulating the PDE as a system of…
Exponential time integrators have been applied successfully in several physics-related differential equations. However, their application in hyperbolic systems with absorbing boundaries, like the ones arising in seismic imaging, still lacks…