Abstract
The paper is devoted to construction and development of new method for numerical solution of hyperbolic type equations [14, 17]. In the previous papers [4, 5, 6, 7, 8, 9] authors have investigated theoretically and tested experimentally 26 different finite‐difference schemes on 4 point patterns for the simplest hyperbolic equation: linear advection equation. This equation has the main features of every hyperbolic equation and is the important part of many mathematical models. In other cases the advection operator is the important part of the full operator of the problem. All 26 schemes have been compared experimentally on the special representative set of tests. Nevertheless to simplicity of the equation, almost all schemes have different disadvantages. They are discussed in detail in the cited papers. So, the investigation of new schemes for this equation is still an important task. In [4, 5, 6, 7, 8, 9] some new schemes were constructed for solving this advection equation. The nonlinear monotone Babenko scheme ("square") proved to be the best among all 26 schemes. So, it is a big interest to generalize this scheme to more difficult equations. The important example is a quasi‐linear advection equation. In this paper our basic aim is to construct a quasi‐monotone nonlinear Babenko scheme for solving the quasi‐linear advection equation and to test it experimentally. The monotonisation of the scheme is done by adding the artificial diffusion with limiters. We also present advanced results of comparative analysis of the new scheme with other known schemes. We have considered explicit and implicit upwind approximation schemes [4, 6, 13, 16] which is firstorder accurate in time and space, the Lax‐Wendroff scheme [4] which is the first order accurate in time and second order accurate in space. We also analyze the monotonised “Cabaret” scheme proposed in [10, 11]. It is second order accurate in time and space, and its monotonisation is based on apriori knowledge of the dependence region of the exact solution. The authors of this scheme called it by “jumping advection”. The considered schemes are compared numerically by using a set of tests, which is similar to one used in [4, 5, 6, 8]. Šiame straipsnyje pasiūlyta kvazi‐monotonie netiesine Babenkos skirtumu schema kvazitiesinei pernešimo lygčiai spresti. Schemos monotoniškumas pasiekiamas pridedant dirbtine difuzija su apribojimais. Pateiktas šios schemos palyginimas su kitomis schemomis. Taip pat analizuojama antros eiles pagal laika ir erdve monotonine “Cabaret” schema. Pateikti testu rezultatai.
Highlights
In the previous papers [4, 5, 6, 7, 8, 9] authors have investigated theoretically and tested experimentally 26 different finite-difference schemes on 4 point patterns for the simplest hyperbolic equation: linear advection equation
In [4, 5, 6, 7, 8, 9] some new schemes were constructed for solving this advection equation
As it was done in [4, 5, 8] the numerical solution of problem (1.1) is obtained for all stated above initial data for following parameters: l = 520, l1 = 0, l2 = 20, T = 1000, h = 1
Summary
With finite (mostly) initial condition u(x, 0) = u0(x). We are going to construct the Babenko scheme for this equation and to test it on the special set of test solutions. The initial functions for this set have following forms: 2(x−l1 l2 −l1. 1, x ∈ [l1, +∞) , They have a form of triangle, rectangle, left-hand triangle, right-hand triangle, stepdown and step-up functions, respectively. Are the exact solutions for initial conditions defined above: 1. X ∈ [l1, l1 + t] , 0 < t < 2(l2 − l1), 1, x ∈ [l1 + t, l2 + 0.5t] , 0 < t < 2(l2 − l1), u(x, t) = 0, x ∈/ [l1, l2 + 0.5t] , 0 < t < 2(l2 − l1), 2(l2 − l1)t , t ≥ 2(l2 − l1), 0, x ∈/ l1, l1 + 2(l2 − l1)t , t ≥ 2(l2 − l1), x−l1 t+(l2 −l1. To compare numerical and exact solutions we use finite-dimensional analogs of norms in spaces C, L1, L2
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