Spectral viscosity approximations to Hamilton--Jacobi solutions (Q2706371)

The spectral viscosity approximate solution of convex Hamilton-Jacobi equations NEWLINE\[NEWLINE\partial_t u(x,t)+ H\bigl(x,t, \nabla_xu(x,t) \bigr)=0\text{ in }\Omega \times[0,T],\;0<T <\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)=\varphi (x)\text{ in }\Omega,\;\varphi\in L^\infty (\Omega),NEWLINE\]NEWLINE where the initial condition \(\varphi(x)\) is \(2\pi\)-periodic in \(x\), and the Hamiltonian \(H(x,t,p)\) is strictly convex with respect to \(x\) and \(p\), and \(2\pi\)-periodic in \(x\), is studied. The author proved (by a spectral viscosity method) that the numerical solution converges to the exact unique viscosity solution of Hamilton-Jacobi equation and obtained the \(L^1\)-convergence rate of the order \(1-\varepsilon\), \(\varepsilon>0\).