Lattice Boltzmann Simulations For Solitary Waves Of Nonlinear Patrial Differential Equations

The lattice Boltzmann method has developed into a new numerical method in thelast20years. Now it has been a new powerful computational tool for thecomputational fluid dynamics and systems governed by related partial differentialequations.Unlike conventional numerical schemes based on discretizations ofmacroscopic continuum equations, the lattice Boltzmann method is based onmicroscopic models and mesoscopic kinetic equations. The lattice Boltzmann methodprovides many of the advantages such as simple program code, easy implementationof boundary conditions and so on. In this paper the lattice Boltzmann method is usedto study some partial differential physics equations with the solitary waves.First we reviewed the lattice Bhatnagar-Gross-Krook model, the applications of thelattice Boltzmann method in fluid dynamics and nonlinear partial differentialequations. We also describe the current research status of the lattice Boltzmannsimulation on solitary wave.In Chapter2, the Chapman analysis and the time multiscale analysis are executed tothe lattice Boltzmann equation. A series of partial differential equations in differenttime scales are obtained. We construct the lattice Boltzmann models for theKorteweg-de Vries equation, the Kadomtsev-Petviashvili equation, the nonlinearSchr dinger equation and the solitary waves in space plasma respectively. Toconstruct the lattice Boltzmann model satisfied conservation laws for the Korteweg-deVries equation and the nonlinear Schr dinger equation, we regarded the originalequation and its conservation as equations. The conservation forms of theKorteweg-de Vries equation and the nonlinear Schr dinger equation are givenrespectively. For the Korteweg-de Vries equation we construct a three-conservationmodel. For the nonlinear Schr dinger equation, we extend the lattice Boltzmann theory to the complex domain and construct an complex lattice Boltzmann model. Themodel has the complex partial differencial equations, the complex equilibriumdistribution functions and the complex moments. For the solitary waves in spaceplasma, we construct the lattice Boltzmann models of the unmagnetized plasma andthe beam-plasma system. By using multiple equilibrium distribution functions andselecting the appropriate moments of the equilibrium distribution functions, thecorresponding macroscopic equations can be restored. Finally, the error analysis isdone.Using the model we simulate the solitary waves of the equations such asKorteweg-de Vries equation, the Kadomtsev-Petviashvili equation, the nonlinearSchr dinger equation, the coupled nonlinear Schr dinger equations, theGross–Pitaevskii equation, the generalized Gross–Pitaevskii equation and theion-acoustic solitary waves in unmagnetized plasma, the ion-and electron-acousticsolitary waves in beam-plasma system. The simulations include the single solitonpropagation and the two solitons interaction both in the one-dimensional andtwo-dimensional. The results show the solitary wave phenomenons in these nonlinearsystems. The lattice Boltzmann solutions are consistent with the exact solutions. Thedependency relationship between the error and the grid number is given and theresults show that the models are convergent. By comparing the lattice Boltzmannsolutions with the other numerical solutions, we find the accuracy of thelatticeBoltzmann model is higher than other numerical methods. The conservation latticeBoltzmann models keep the conservation of the conserved quantity. The latticeBoltzmann method is a very effective method to simulate the solitary waves of thenonlinear partial differential equations.