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Spurious solutions for the advection‐diffusion equation using wide stencils for approximating the second derivative
Author(s) -
Frenander Hannes,
Nordström Jan
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22210
Subject(s) - spurious relationship , stencil , truncation error , mathematics , advection , truncation (statistics) , mathematical proof , convergence (economics) , diffusion , stability (learning theory) , derivative (finance) , mathematical analysis , geometry , computer science , physics , statistics , computational science , machine learning , financial economics , economics , thermodynamics , economic growth
A one‐dimensional steady‐state advection‐diffusion problem using summation‐by‐parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes are equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments.