Abstract
The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been considered. In this paper, we propose high-order numerical methods to solve the Klein-Gordon equation, present the energy and linear momentum conservation properties of our numerical schemes, and show the optimal error estimates and superconvergence property. We also verify the performance of our numerical schemes by some numerical examples.
Highlights
The Klein-Gordon equation is a widely used model in relativistic quantum mechanics
We propose high-order local discontinuous Galerkin (LDG) methods that have provable properties of energy and linear momentum conservation, optimal convergence and superconvergence
We propose new methods for the Klein-Gordon equation based on the local discontinuous Galerkin methods because LDG methods have some advantage of classical finite element methods
Summary
The Klein-Gordon equation is a widely used model in relativistic quantum mechanics. It is used to describe the free particle wave function. In [3], a finite difference method along with an operator splitting method was proposed to solve the equation on an unbounded domain All of these finite difference schemes have second-order convergence in space and time. We propose high-order local discontinuous Galerkin (LDG) methods that have provable properties of energy and linear momentum conservation, optimal convergence and superconvergence. We propose new methods for the Klein-Gordon equation based on the local discontinuous Galerkin methods because LDG methods have some advantage of classical finite element methods. We can show that our numerical results are consistent with our theoretical results
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