Related papers: A general framework for deriving integral preservi…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the…
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…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
The first part of this paper introduces sufficient conditions to determine conservation laws of diffusion equations of arbitrary fractional order in time. Numerical methods that satisfy a discrete analogue of these conditions have…
Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important…
This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential-algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential-algebraic…
We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on…
In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…
Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time-dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can…
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…
In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods…