For the splitting method of the nonlinear heat equation with initial datum in W^1,q
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In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in Ω⊂ℝ^d (d≥ 1): ∂_t u = Δ u + λ |u|^p-1 u in Ω×(0,∞), u=0 in ∂Ω×(0,∞), u ( x,0) =ϕ ( x) in Ω. where λ∈{-1,1} and ϕ∈ W^1,q(Ω)∩ L^∞ (Ω) with 2≤ p < ∞ and d(p-1)/2<q<∞. We establish the well-posedness of the approximation of u in L^r-space (r≥ q), and furthermore, we derive its convergence rate of order 𝒪(τ) for a time step τ>0. Finally, we give some numerical examples to confirm the reliability of the analyzed result.
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