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On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms. (English) Zbl 1059.70012
This work is devoted to numerical solution of the following non-homogeneous Hamilton-Jacobi equation \begin{aligned} u_t+H(t,x,u,Du)=G(t,x,u)&\quad\text{in } \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x)&\quad\text{in } \mathbb{R}^N. \end{aligned}\tag{1} Let $$u$$ be the unique viscosity solution of (1). The authors split the problem into the ordinary differential equation \begin{aligned} v_t=G(t,x,v)&\quad\text{in }\mathbb{R}^N\times(s,T),\\ v(x,s)=v_0(x)&\quad\text{in } \mathbb{R}, \end{aligned} \tag{2} and into the homogeneous Hamilton-Jacobi equation \begin{aligned} v_t+H(t,x,v,Dv)=0&\quad\text{in } \mathbb{R}^N\times(s,T),\\ v(x,0)=v_0(x)&\quad\text{in } \mathbb{R}^N. \end{aligned} \tag{3} Denoting by $$E(t,s)$$ and $$S(t,s)$$ the solution operators for (2) and (3) respectively, they show that the functions $v(x,t_i)=S(t_i,t_{i-1})E(t_i,t_{i-1})v(\cdot,t_{i-1})(x),\quad v(x,0)=v_0(x),$ uniformly converge to $$u$$ with a rate of convergence which only linearly depends on $$\| u_0-v_0\|$$ and on the time step $$t_i-t_{i-1}\equiv\Delta t$$. In particular, the solution operator $$E$$ may be Euler operator as well as an exact solution operator. The operator $$S$$ is instead the viscosity solution operator. The obtained linear rate of convergence improves previous results.

##### MSC:
 70H20 Hamilton-Jacobi equations in mechanics 70-08 Computational methods for problems pertaining to mechanics of particles and systems
##### Keywords:
Euler operator; viscosity solution operator; front tracking
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