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An explicit stable numerical scheme for the $$1D$$ transport equation. (English) Zbl 1244.65131
Summary: We derive 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.

MSC:
 65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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