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S25 Métodos Numéricos |
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Title:
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A MATHEMATICA PROGRAM TO EVALUATE THE TRUNCATION ERROR
ARISING FROM THE DISCRETIZATION OF PARTIAL DIFFERENTIAL EQUATIONS
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Summary :
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ABSTRACT: THE MODIFIED EQUATION, INTRODUCED BY HIRT IN THE LATE SIXTIES, STILL IS ONE OF THE MOST IMPORTANT TOOLS FOR EVALUATING THE TRUNCATION ERROR TERMS ARISING FROM THE DISCRETIZATION OF PARTIAL DIFFERENTIAL EQUATIONS. A FEW YEARS LATER, WARMING AND HYETT PLACED THE MODIFIED EQUATION IN A SOLID MATHEMATICAL BASIS. HOWEVER, THE BASIC CHARACTERISTIC OF THE ALGORITHM, NAMELY THAT IT REQUIRES EXTENSIVE ALGEBRAIC MANIPULATIONS OF THE DISCRETIZED EQUATION, REMAINED UNCHANGED. THE REPETITIVE OPERATIONS PERFORMED BY THE ALGORITHM MAKE IT A GOOD CANDIDATE FOR AUTOMATION USING A COMPUTER. BUT SINCE THESE OPERATIONS REQUIRE SYMBOLIC MANIPULATION AND DIFFERENTIATION, THE ALGORITHM IS UNSUITED FOR IMPLEMENTATION IN "STANDARD" PROGRAMMING LANGUAGES SUCH AS PASCAL, FORTRAN OR C. FOR THIS PARTICULAR ALGORITHM, APPROPRIATE PROGRAMMING LANGUAGES ARE THOSE CAPABLE OF PERFORMING SUCH SYMBOLIC MANIPULATIONS. ONE SUCH LANGUAGE COMES WITH THE SOFTWARE PACKAGE MATHEMATICA.
THIS WORK DESCRIBES EQMOD, AN IMPLEMENTATION OF HIRT S MODIFIED EQUATION ALGORITHM. EQMOD IS WRITTEN IN MATHEMATICA S NATIVE PROGRAMMING LANGUAGE. SOME SIMPLE EXAMPLES ARE PRESENTED TO DEMONSTRATE EQMOD S CAPABILITIES.
KEYWORDS: EQMOD, FINITE DIFFERENCES EQUATION, PARTIAL DIFFERENTIAL EQUATIONS, TRUNCATION ERROR, MODIFIED EQUATION.
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Author :
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Fortuna, Armando de O.
Rolnik, Vanessa Portioli
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Paper View :
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