THE ACCURATE ELEMENT METHOD: A NEW PARADIGM FOR NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

Transcription

1 THE PUBISHING HOUSE PROCEEDINGS O THE ROMANIAN ACADEMY, Series A, O THE ROMANIAN ACADEMY Volume, Number /, pp. 6 THE ACCURATE EEMENT METHOD: A NEW PARADIGM OR NUMERICA SOUTION O ORDINARY DIERENTIA EQUATIONS Maty BUMENED *, Paul Cizmas ** * Politehnica University Bucharest Bucharest, Romania ** Teas A&M University College Station, Teas, USA Corresponing author: Maty BUMENED, mblum@itcnet.ro This paper presents the evelopment of a new methoology for numerically solving orinary ifferential equations. This methoology, which we call the accurate element metho AEM, breaks the connection between the solution accuracy an the number of unknowns. The ifferential equations are iscretize by using finite elements, like in the finite element metho EM. A key feature of the AEM is the methoology evelope for eliminating unknowns insie the element by using the relations provie by the governing equations. The results of the AEM applie to a secon-orer orinary ifferential equation ODE show that the orer of the local approimation function oes not affect the number of unknowns use to iscretize the ifferential equation. This is a breakthrough in the paraigm use to approach the numerical solution of ifferential equations. or a secon-orer ODE it is shown that the iscretize solution has the same number of unknowns, whether the local approimation or interpolation function is a thir-egree or a nineteen-egree polynomial. The implication of this result is that higher-egree interpolation functions can be use without increasing the number of unknowns. or a specifie accuracy, the usage of elements with high-egree interpolation functions leas to a reuction in the necessary number of elements. It is shown herein that for the solution of a secon-orer ODE, two AEM elements with seventh-egree polynomial interpolation functions generate a more accurate solution than EM elements with a linear interpolation function. Since in this case the number of unknowns per element is the same for both the AEM an the EM elements, this results in a 5-fol reuction in the number of unknowns. This reuction of the number of unknowns leas to a reuction of the computational time. Key wors: Orinary ifferential equations; inite element metho.. INTRODUCTION In recent years, various numerical methos have been evelope for solving orinary ifferential equations an partial ifferential equations. The most general an wiely use methos inclue the finite ifference metho, finite volume metho an finite element metho EM. The common goal of these methos is to approimate as well as possible the solution of the governing equations escribing various physical phenomena. Several options are currently use to improve the accuracy of these methos: increasing the egree of the polynomials that locally approimate the solution, that is, p-type refinement; ecreasing the size of the elements which iscretize the computational omain in orer to reuce the variation of the unknowns over the element, that is, h-type refinement; using anisotropic elements, that is, having ifferent orers of epansion for each spatial irection in an element; using non-conforming elements with either h-type or p-type non-conformities; 5 using aaptive refinement strategies. or all the methos currently use to solve ifferential equations, the outcome of increasing solution accuracy is an increase in the number of unknowns. To minimize the number of unknowns, each metho tries to optimize the relation between the orer of the approimation function an the size of the element, such as in the h-p aaptive finite element or spectral element methos. This paper presents the evelopment of a new methoology for the numerical solution of ifferential equations. This methoology, which we call the accurate element metho AEM, breaks the connection between the solution accuracy an the number of

2 Maty Blumenfel, Paul Cizmas unknowns. Consequently, the number of unknowns corresponing to high-accuracy solutions obtaine using high-orer approimation polynomials is equal to the number of unknowns corresponing to low-orer approimation polynomials. As a result of the reuction of the number of unknowns, the computational time ecreases. The AEM is escribe in the net section. The general escription of the AEM is followe by the presentation of the methoology. A secon-orer, linear ODE is use to illustrate the implementation of the AEM.. GENERA DESCRIPTION O THE ACCURATE EEMENT METHOD The AEM will be presente by comparing it to the EM. The strategies of both the AEM an the EM can be summarize by the following three steps: iscretization of the omain D on which the governing equations must be integrate; local approimation of the solution of the governing equations, in which the information given by each element is concentrate at the noes of the element; an reconstruction of the omain D, obtaine by writing noal equations that bring together the information given by the elements ajacent to each noe. The solution of the system of equations obtaine in this way represents the noal unknowns. The first an thir steps of the AEM are ientical to those of the EM. The two methos iffer only in how the secon step of their common strategy is accomplishe. Note that the secon step is crucial in obtaining an accurate solution of the governing equations. As will be shown herein, the AEM uses a much smaller number of elements than the EM to obtain a similar accuracy. In step two, the local approimation of the EM solution is one by using interpolation functions I. Usually these interpolation functions are low egree polynomials that lea to poor approimations. Consequently, the number of elements use in the EM must be large in orer to obtain a result with reasonable accuracy. The local approimation in the AEM will be one by high-egree polynomial functions. The high-egree polynomial functions will be calle concorant functions C. Because the functions use for local approimation are high-egree polynomials, only a small number of elements is neee to obtain an accurate solution in the AEM. The eample presente below shows that the accuracy obtaine in the AEM by using two elements with a seventh-egree polynomial C is better than the accuracy obtaine in the EM by using elements with a linear I. The egree of the polynomials use in the AEM is usually higher than seven, since, as will be shown herein, the computational effort in the AEM is not affecte by the egree of the polynomial. This is an important ifference between the AEM an the EM. High-egree polynomials may be use for the I in the EM, but this increases the number of unknowns of the iscretize system of equations. In the AEM, the egree of the polynomial use in the local approimation oes not affect the number of unknowns of the iscretize system of equations. This is an important avantage of the AEM compare to the finite element, finite volume or finite ifference methos. METHODOOGY AND EXAMPE To introuce the methoology of the AEM, let us start by presenting a simple but yet meaningful case, the solution of a linear, secon-orer orinary ifferential equation : with the bounary conitions The analytical solution of this equation with the specifie bounary conitions is: et us consier a C that is a seventh-egree polynomial: an let us write using a natural coorinate η/, is the length of the element. Equation can be written in vectorial form using the natural coorinate η:

3 The accurate element metho: a new paraigm for numerical solution of orinary ifferential equations 5 5 an The values of the function an its erivatives at both ens of the element can be written as: 6 or 7 The notation an with k has been use in equation 6. The inverse of matri A is 8 an B can be rearrange as:

4 6 Maty Blumenfel, Paul Cizmas. B 9 The vector of coefficients C can be written using equations 7, 8 an 9 as: B A C , 5 5 8, , et us erive a relation similar to but more general than the stiffness matri relation. This relation shoul provie a link between the variables i an the variable erivatives i k, k. This relation, which is a generalization of the stiffness matri relation, will be calle the complete transfer relation CTR. The erivation of the CTR can be one base on the physical phenomenon that is escribe by the governing equations or by using the Galerkin metho. The latter approach will be use herein, because of its generality. et us use the following weighting function N w in the Galerkin metho, an let us rewrite the governing equation in a more general form:. c Using the Galerkin projection one obtains N T c N T N T. After integrating by parts the first term, equation becomes c et us now replace the function of the thir term of the left-han sie by the epression given in equation 5. Equation becomes

5 5 The accurate element metho: a new paraigm for numerical solution of orinary ifferential equations 7 Note that often the thir term of equation is not accounte for while eriving the stiffness matri in the EM. Even when this term is accounte for, the function in the integral is approimate by a low egree polynomial. This term is responsible for introucing errors in the EM. Unlike the EM, the AEM uses a high orer polynomial to approimate. The result of this approach is that equation has 8 unknowns. There are, however, only available relations: equations an bounary conitions. Consequently, aitional relations are necessary. Usually, when more information is neee, this information is obtaine either from some relations with other elements or by accepting a reasonable approimation. Neither of these options is use in the AEM. Instea, the necessary information is obtaine accurately from insie the element, that is, from the governing equation itself. Consequently, the secon an thir erivatives are an 5 6 After substituting equations 5 an 6 in equation one obtains The unknowns an, which we call the apparent unknowns, have been eliminate by using the governing equations. Similar equations can be obtaine by using higher orer Cs, as shown below: - seventh-egree concorant function, C8 - nineteen-egree concorant function, C

6 8 Maty Blumenfel, Paul Cizmas 6 or comparison, let us show the EM solution obtaine using a linear interpolation function I Table shows the variation of as a function of the egree of the polynomial in the C. This solution is base on one element. or comparison, the value of calculate using the EM with an I is also inclue. The eact value is obtaine from the solution as The variation of the error shows that an accurate solution can be obtaine with only one AEM element with C. Error % I C C e- C e-7 C e- C e- Table : Solution base on one element. The variation of as a function of the number of elements is shown in Table. Results are shown for the EM with I an the AEM with C an C8. The avantage of using the AEM an high-egree polynomials is obvious. Two AEM elements with C8 prouce more accurate results than EM elements with I. our AEM elements with C8 prouce a solution with an error of the orer of -8 %. I C C8 No. Elem. Error % Error % Error % e e- -.7e e- -.8e e- * e- * e-5 * e e-7 * e- * * * Table : Variation of the value of as a function of the number of elements an egree of the C/I.. Conclusions A new metho, which we call the accurate element metho, has been presente herein. This new methoology can be applie to numerically solve both orinary an partial ifferential equations. Herein the

7 7 The accurate element metho: a new paraigm for numerical solution of orinary ifferential equations 9 AEM has been applie to solve orinary ifferential equations. A key feature of the AEM was the methoology evelope for eliminating unknowns insie the element by using the relations provie by the governing equations. The results of the AEM applie to a secon-orer ODE showe that the orer of the local approimation function i not affect the number of unknowns use to iscretize the ifferential equation. This is a breakthrough in the paraigm use to approach the numerical solution of ifferential equations. or a secon-orer ODE it was shown that the iscretize solution has the same number of unknowns, whether the local approimation or interpolation function is a thir-egree or a nineteen-egree polynomial. The implication of this result is that higher-egree interpolation functions can be use without increasing the number of unknowns. or a specifie accuracy, the usage of elements with high-egree interpolation functions leas to a reuction in the necessary number of elements. It was shown herein that for the solution of a secon-orer ODE, two AEM elements with seventh-egree polynomial interpolation functions generate a more accurate solution than EM elements with a linear interpolation function. Since in this case the number of unknowns per element was the same for both the AEM an the EM elements, this resulte in a 5-fol reuction in the number of unknowns. REERENCES. M. BUMENED. A New Metho for Accurate Solving of Orinary Differential Equations. Eitura Tehnica, Bucharest,.