On the mesh for difference schemes of higher accuracy for the heat-conduction equation

In domain Ω = [ 0 , 1 ] × ℝ 0 + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq838.tif"/> we consider the initial-boundary value problem for the heat-conduction equation ∂ w ∂ t = ∂ 2 w ∂ x 2 ;   x ∈ ( 0 , 1 ) , t > 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq839.tif"/>