An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives

by   Qi Tao, et al.

In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of local discontinuous Galerkin (LDG) method and ultra-weak discontinuous Galerkin (UWDG) method. Firstly, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can then use the discrete Sobolev and Poincaré inequalities to obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.


page 1

page 2

page 3

page 4


A Local Deep Learning Method for Solving High Order Partial Differential Equations

At present, deep learning based methods are being employed to resolve th...

Two-derivative deferred correction time discretization for the discontinuous Galerkin method

In this paper, we use an implicit two-derivative deferred correction tim...

A discontinuous Galerkin method for nonlinear biharmonic Schrödinger equations

This paper proposes and analyzes an ultra-weak local discontinuous Galer...

A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

This paper develops a unified general framework for designing convergent...

Post-processing and improved error estimates of numerical methods for evolutionary systems

We consider evolutionary systems, i.e. systems of linear partial differe...

On optimal recovering high order partial derivatives of bivariate functions

The problem of recovering partial derivatives of high orders of bivariat...

Three discontinuous Galerkin methods for one- and two-dimensional nonlinear Dirac equations with a scalar self-interaction

This paper develops three high-order accurate discontinuous Galerkin (DG...

Please sign up or login with your details

Forgot password? Click here to reset