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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.

70H20 Hamilton-Jacobi equations in mechanics
70-08 Computational methods for problems pertaining to mechanics of particles and systems
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