On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms (Q2719219)

This work is devoted to numerical solution of the following non-homogeneous Hamilton-Jacobi equation NEWLINE\[NEWLINE\begin{aligned} u_t+H(t,x,u,Du)=G(t,x,u)&\quad\text{in } \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x)&\quad\text{in } \mathbb{R}^N. \end{aligned}\tag{1}NEWLINE\]NEWLINE Let \(u\) be the unique viscosity solution of (1). The authors split the problem into the ordinary differential equation NEWLINE\[NEWLINE\begin{aligned} v_t=G(t,x,v)&\quad\text{in }\mathbb{R}^N\times(s,T),\\ v(x,s)=v_0(x)&\quad\text{in } \mathbb{R}, \end{aligned} \tag{2}NEWLINE\]NEWLINE and into the homogeneous Hamilton-Jacobi equation NEWLINE\[NEWLINE\begin{aligned} v_t+H(t,x,v,Dv)=0&\quad\text{in } \mathbb{R}^N\times(s,T),\\ v(x,0)=v_0(x)&\quad\text{in } \mathbb{R}^N. \end{aligned} \tag{3}NEWLINE\]NEWLINE Denoting by \(E(t,s)\) and \(S(t,s)\) the solution operators for (2) and (3) respectively, they show that the functions NEWLINE\[NEWLINE v(x,t_i)=S(t_i,t_{i-1})E(t_i,t_{i-1})v(\cdot,t_{i-1})(x),\quad v(x,0)=v_0(x), NEWLINE\]NEWLINE 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.