求解对流扩散方程的Haar小波方法(英文).docx

求解对流扩散方程的Haar小波方法(英文)
HaarWaveletMethodforSolvingConvection-DiffusionEquation
Introduction
Theconvection-diffusionequationisafundamentalpartialdifferentialequationthatarisesfrequentlyinphysicalandengineeringapplications.Theequationcombinesboththeeffectsofdiffusionandadvection,whichmakesitssolutionachallengingtask.Severalnumericalmethodshavebeendevelopedtosolvetheconvection-diffusionequation.Inthispaper,wepresenttheHaarwaveletmethodforsolvingtheconvection-diffusionequation.
HaarWaveletMethod
Haarwaveletsaremathematicalfunctionsthatbelongtothefamilyoforthogonalwavelets.Theyareusedinsignalprocessingandnumericalanalysisfortheirexcellentpropertiessuchasorthogonality,compactsupport,andmultiresolution.Haarwaveletisdefinedasfollows:
(1)ψ(x)={1for0≤x<1/2;-1for1/2≤x<1;0otherwise}
TheHaarwaveletfunctioncanbetranslatedandscaledasfollows:
(2)ψ(a,b)(x)=(1/√a)ψ((x-b)/a)
Here,aandbarescalingandtranslationparameters,respectively.Scalingparameteraisusedtochangethefrequencyofthewavelet,whilethetranslationparameterbisusedtoshiftthewaveletalongthex-axis.
Tosolvetheconvection-diffusionequation,wefirstdiscretizeitusingafinitedifferencemethod.Then,weusetheHaarwaveletmethodtosolvetheresultingsystemofequations.Letusconsidertheone-dimensionalconvection-diffusionequation:
(3)∂u/∂t+c∂u/∂x=D∂^2u/∂x^2
Here,u(x,t)isthedependentvariable,cistheconvectioncoefficient,Disthediffusioncoefficient,andtisthetime.Wediscretizetheequationusingafinitedifferencemethodasfollows:
(4)(u_i^(n+1)-u_i^n)/Δt+c(u_i^n-u_{i-1}^n)/Δx=D(u_{i+1}^n-2u_i^n+u_{i-1}^n)/Δx^2
Here,iisthespatialindex,andnisthetimestep.
Then,weapplytheHaarwaveletmethodbyexpressingthesolutionu(x,t)asalinearcombinationofHaarwaveletsasfollows:
(5)u(x,t)=∑_{i,j=0}^{2^N-1}u_{i,j}^nψ((x-i2^kΔx)/2^k)
Here,u_{i,j}^nisthecoefficientoftheHaarwaveletfunctionψ((x-i2^kΔx)/2^k)atthei-thspatiallocationandj-thresolutionlevelatthen-thtimestep.Nisthenumberofspatialdiscretizationpoints,kistheresolutionlevel,andΔxisthespatialstepsize.
Substitutingequation(5)intoequati