An accelerated splitting-up method for parabolic equations

الأصل · EN

We approximate the solution u of the Cauchy problem ∂/∂ t u(t,x)=Lu(t,x)+f(t,x), (t,x)∈(0,T]×ᵈ, u(0,x)=u₀(x), x∈ᵈ by splitting the equation into the system ∂/∂ t vᵣ(t,x)=Lᵣvᵣ(t,x)+fᵣ(t,x), r=1,2,...,d₁, where L,Lᵣ are second order differential operators, f, fᵣ are functions of t,x, such that L=∑ᵣ Lᵣ, f=∑ᵣ fᵣ. Under natural conditions on solvability in the Sobolev spaces Wᵐₚ, we show that for any k>1 one can approximate the solution u with an error of order δᵏ, by an appropriate combination of the solutions vᵣ along a sequence of time discretization, where δ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [7], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of δ.