Abstract
We derive in this paper a numerical scheme in order to calculate solutions of $1D$ transport equations. This $2nd$-order scheme is based on the method of characteristics and consists of two steps: the first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two $2nd$-order interpolation schemes so as to preserve the maximum principle. The resulting method is designed for classical solutions and is unconditionally stable.
Highlights
The method of characteristics has been used for 40 years as a theoretical tool to prove existence of smooth solutions to the linear transport equation
We shall present numerical results for the linear transport equation and the Burgers equation. This model is designed for dimension 1 and for smooth solutions to the transport equation. It is based on properties of the characteristic flow that we use in the approximation of the foot of the characteristic curve (2nd-order scheme) and on geometric considerations in the interpolation step so as to ensure the maximum principle
We have designed in this paper a new numerical method of characteristics which is 2nd-order accurate so as to simulate smooth solutions to convection problems such as the linear transport equation or the Burgers equation
Summary
The method of characteristics has been used for 40 years as a theoretical tool to prove existence of smooth solutions to the linear transport equation. The method consists in two steps: the construction of the characteristic to provide the foot X (t;t + ∆t,x) of the curve passing through x at time t + ∆t (as well as other values required by the integration formula) and the evaluation of the computed solution at time t and position X (t;t + ∆t,x). This model is designed for dimension 1 and for smooth solutions to the transport equation It is based on properties of the characteristic flow that we use in the approximation of the foot of the characteristic curve (2nd-order scheme) and on geometric considerations in the interpolation step so as to ensure the maximum principle
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