The Finite Volume Method For Conservation Laws

Most of partial differential equation arising from physical or engineering science can be formulated into conservation form:It directly reflects conservation laws in natural sciences.From viewpoints of fluid dynamics,it can be obtained from the mass,momentum,energy conservation laws.Because the form (0.2.1) has no other terms such as dispersion,diffusion(caused by nonuniformity of some physical states) ,reaction,memory,damping and relaxation etc, smoothness of solution of (0.2.1) may be loss as times goes on. Even for the smooth inital data, solutions of (0.2.1) become discontinuous in a finite time. This feature reflects the physical phenomenon of breaking of waves and development of shock waves.In the fields of fulid dynamics,(0.2.1) is an approximation of small visvosity phenomenon.If viscosity(or the diffusion term, two derivatives) are added to (0.2.1),it can be researched in the classical way which say that the solutions become very smooth immediately even for coarse inital data because of the diffusion of viscosity.A natural idea(method of regularity) is obtained as follows: solutions of the viscous convection-diffusion problem approachs to the solutions of (0.2.1)when the viscosity goes to zeros.Another method is numerical method such as difference methods,finite element method,spectrum method or finite volume method etc.Numerical solutions which is constructed from the numerical scheme approximate to the solutions of the hyperbolic con-ervation laws(0.2.1) as the discretation parameter goes to zero.The aim of these two methods is to construct approximate solutions and then to conside the stability of approximate so-lutions(i,e.the upper bound of approximate solutions in the suitable norms,especally for that independent of the approximate parameters).Using the compactness framework(such as BV compactness,L1 compactness and compensated compactness etc) and the fact that the truncation is small,the approximate function consquence approch to a function which is exactly the solutions of (0.2.1) in some sense of definiton. Due to the poor regularity of solutions at large time.(0.2.1) can not defined in classical way.i,e.,the definition of the derivatives at any points has no sense. So it may be rather difficult in the research of classical way and must be defined in weak sense.In order to guarantee the uniqueness of weak solutions, a condition (entropy inequality) must be need to pick out 'good' solution (entropy solutions). In the fields of fluid dynamics, entropy inequality reflects the second law of thermodynamics.i.e..entropy must increase across shock waves(a kind of discontinuity).All kind of approximate schemes should reflect the fact that it must satisfies some kindof discrete entropy inequality) .From the view of practical computation,stability and theo-retical error of any kind discrete schemes all dependend of the smoothness of the solution of (0.2.1).Generally,the approximate solution have good stability and theoretial error in the area where the solutions have more regularity and poor stability and theoretial error in other area. For the practical computation of (0.2.1),usual approximate schemes has poor computation results in the area of poor regularity such as discontinuities.Firstly,the location of areas of poor regularity is rather difficult.Secondly,approximation in such area is poor.For example,simulation of physical ossciation is a difficult task. All facts reflects that the lackness of smoothness of solution causes difficulties in numericall research.Double variable technique is used by Kruzkov in 70's to obtain the existence,uniqueness and regularity of entropy solution to (0.2.1) for the scalar case,especially the contractive properity of entropy solution.Kuznetsov applied this technique to approximation of scalar hyperbolic conservation laws (0.2.1) in 1976. A parabolic regularity(viscosity method) and difference schemes in the rectangle mesh is constructed and convergence rata is obtained in the norm L1.His approximation method is lately called Kuznetsov approximation t...