High-Accuracy Difference Schemes for the Nonlinear Transfer Equation

Abstract

In the present paper, for the initial boundary value problem for the non‐homogeneous nonlinear transport equationthe basic principles for constructing difference schemes of any order of accuracy O(#GTM), M ≥ 1, on characteristic grids with the minimal stencil were introduced. To construct a difference scheme the Steklov averaging idea for the right‐hand sidewas used. The case of f(u) = λu2 was investigated in detail. A strict analysis of the order of approximation, stability, and convergence in nonlinear case was made. The performed numerical experiments justify theoretical results.

Exact difference schemes and schemes of higher order of approximation for convection-diffusion equation. I

“…with the initial and boundary conditions (5). The difference scheme approximating the above problem is [8]…”

“…We consider the function u(x, t) = sin 2 (π(x − t))/(1 − t sin 2 (π(x − t))) (see Fig. 1) that satisfies the problem (17), (5) in the domain Q T = [0, 1] × [0, 0.9] with a = 1, f 1 ≡ 1 and f 2 (u) = u 2 [15]. The corresponding difference scheme is Table 1 presents the numerical results for time and space steps satisfying the condition γ = 1 and demonstrates the exactness of the difference scheme.…”

Difference schemes of arbitrary order of accuracy for semilinear parabolic equations

“…where constants c p > 0. Then difference scheme (13)-(14) approximates problem (9)- (10) with the truncation error equal to O τ m 2 .…”

“…It is worth mentioning here the paper by Mickens [9] in which nonstandard finite difference schemes are introduced. In [10] the investigations of the order of approximation, stability, and convergence of the high accuracy difference schemes for the nonlinear transfer equation ∂u ∂t + u ∂u ∂x = f (u) have been made. The EDS and the difference schemes of an arbitrary order of approximation for the parabolic equations with travelling wave solutions u(x, t) = U (x − at) were constructed in [7,8].…”