Related papers: Runge-Kutta Physics Informed Neural Networks: Form…
We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations (DAEs) w.r.t. the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive…
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…
In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta…
Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical…
The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a…
We apply exponential Runge Kutta time discretizations to semilinear evolution equations $\frac { {\rm d} U}{{\rm d} t}=AU+B(U)$ posed on a Hilbert space ${\mathcal Y}$. Here $A$ is normal and generates a strongly continuous semigroup, and…
This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and…
Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…
In this paper, we develop a new type of Runge--Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple…
In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of…
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…
In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…
This study presents a numerical analysis of the Field-Noyes reaction-diffusion model with nonsmooth initial data, employing a linear Galerkin finite element method for spatial discretization and a second-order exponential Runge-Kutta scheme…
The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong…
The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\epsilon$, by solving only $$O\Big(\log N \log \frac1\epsilon \Big) $$ linear…
Extended Stability Runge-Kutta (ESRK) methods are crucial for solving large-scale computational problems in science and engineering, including weather forecasting, aerodynamic analysis, and complex biological modelling. However, balancing…
This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…
In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods of stochastic Hamiltonian…
In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell's equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new…
Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for…