A High-accuracy High Precision Direct Algorithm To Solve Non-homogeneous Autonomy System With Exponent Stimulus

Since Zhong Wanxie, a well-known academician proposed the HighPrecision Direct(HPD) integration scheme in1994, this scheme, which isthe academic intersect of computational mechanics, engineeringapplication, computational mathematics has developed quickly andbecome an academic focus. Based on the existing research achievements,this paper focuses on the technology of polynomial expansion, mainly theChebyshev orthogonal polynomial and puts the HPD on a higher leveland carries on a series of researches about long-term-effective HPD withExponent Stimulus.This article has mainly four innovations as following:（1）A new extrapolation of HPD (namely: NW-HPD) and thedefinition of the initial N number.This article propose a new extrapolation of HPD with linearcombination of transfer matrix H coming from different N number byanalyzing the source of error. It also gives the high order interpolationbased on one order interpolation. This interpolation is different fromtraditional interpolation: its linear combination is for the transfer matrix H, not the result of X. So this interpolation is called NW-HPD. Thenumerical example displays that accuracy of NW-HPD enhance muchcompared with HPD.Let the relative error that originate from error term be smaller thanthe given maximum relative error, then we get theorem of the definitionof initial N number. The numerical example displays it can just satisfy thegiven accuracy. So this N number is fine.（2）Expansion with Chebyshev orthogonal polynomials of thesecond kind with Exponent Stimulus.This article devises a high-accuracy high precision directalgorithm to solve non-homogeneous autonomy system with exponentstimulus which is called HHPD-SC. The algorithm expands the stimulusfunction on the right with Chebyshev orthogonal polynomials of thesecond kind with Exponent Stimulus. The article gives a specific methodto construct HHPD-SC, and analyses its accuracy and calculated amountcompared with RK method. The theory and numerical examples displaythat HHPD-SC’s accuracy is higher and its calculated amount is lessthan RK method. What’s more, HHPD-SC is long-term-effectivealgorithm, namely the transfer matrix H is “calculated once, used forever”when the step size is fixed.（3）TB-HHPD-SCwe often meet the right stimulus of measuring kind in practical application, numerical analysis theory emphasizes using piecewise lowdegree polynomial to interpolation approach. Thus this article devises anew long term-effective expanding type HPD namely TB-HHPD-SC,which can be parallel calculated to deal with right piecewise low degreepolynomial stimulus. Theory and numerical examples display: thealgorithm of this text is very effective.（4）PS-HHPD-SCWhen A is Hamilton system matrix, we combine the HHPD-SC andsymplectic algorithm of Hamilton system considering the fine property ofsymplectic algorithm, then we devise PS-HHPD-SC. Theory andnumerical examples show: PS-HHPD-C is better than HHPD-SC andR-K method.