有限体积方法在守恒律中的应用
摘要
来自自然科学与工程领域中的大多数微分方程在数学上表现为守恒形式：它是自然界中的守恒定律在数学上的直接反映，对流体力学方程组而言，它就是质量，动量，能量守恒得到的方程。由于双曲守恒律(0.1.1)没有其它项，如色散(dispersion)，扩散(diffusion)(某物理量分布不均匀引起的输运)，反应(reaction)，记忆(memory)，阻尼(damping)及松弛(relaxation)(描述非平衡态)等，而仅有输运或对流项(convection)(由于流体的流动引起的输运)时，守恒律(0.1.1)的解失去光滑性(这里不特殊说明守恒律就指该意义下)，甚至即使光滑的初始数据，解随着时间的发展会变成不连续，这在物理上表现为激波的形成。从流体力学的角度上看，(0.1.1)事实上就是粘性很小的近似。当考虑粘性后，即在数学上反映为(0.1.1)中多了扩散项(二阶导数项)，即使很粗糙的初始数据，解在瞬间内变的很光滑，这由于流体的粘性扩散引起，这种对流-扩散问题可用古典的微分方程来研究。自然的想法就是当粘性趋于零时，带粘性的对流-扩散问题的解在某意义下趋于无粘性问题(0.1.1)的解，这就是正则化方法。另一办法从离散(数值)角度上研究仅有对流项的守恒律(0.1.1)，如构造它的差分格式，甚至更一般的有限体积格式，有限元及谱方法等，从这些格式构造近似解(常表现为分片多项式)来逼近原守恒律的解。正则化方法与数值角度研究都在于估计解的构造，其次考虑估计解的稳定性，即在适当度量下估计解大小的上界，而且着重研究与正则化系数(如粘性大小)与数值的离散参数(如时间，空间步长大小)无关的上界。在利用紧框架(如BV紧，L~1紧，补偿紧等)得到估计解有子列在某种模意义下收敛到某一个函数，再利用估计格式(估计解所满足的方程)逼近守恒律来证明所得到的函数就是在某种意义下守恒律的解。由于大时间范围内守恒律(0.1.1)的解表现为很差的正则性，它不能在古典意义下定义，即在每一点下的导数无意义，使得古典办法研究遇到很大困难，它只能在弱意义下定义弱解，但往往这种弱解不唯一，需要某条件限制确保解的唯一性，在数学上称为熵条件，满足该条件的弱解称为熵解。从流体力学来看，它事实上是热力学第二定理的反映，即熵越过激波(一种间断)要增加。各种估计格式构造的估计解应反映这一事实，即满足熵不等式。从实际计算来看，总是通过离散化求解，不考虑计算的积累误差，它的稳定性与计算精度都依赖与真解的光滑性，一般说，在解较光滑的区域有较好的稳定性与计算精度，而在较粗糙的区域则相反。对守恒律(0.1.1)的求解来说，一般保持一定精度的数值格式在光滑性较差的区域计算效果不大好，首先，捕捉解的光滑性较差的区域(间断处)较差，
其次，在该区域附近通近不大好，如不能模拟物理上的展荡.所有这些都反映了解的正
则性的缺乏给数值研究带来困难.
Kruzkov【481在70年代用双变t技巧(Double variable teehnique)解决了多维单个双
曲守恒律(0 .1 .1)的摘解的适定性问题，即嫡解的存在性，唯一性及正规性结果，特别摘解
的Ll收缩性质.1976年Kuznetsov[sl]把双变t技巧用于双曲守恒律的估计问题，用抛
物正规(粘性方法)或用R‘上的矩形网格构造差分格式，并得到了相应的L‘收敛速度估
计.他的估计方法常称为Kuznetsov估计理论.1995年Kuznetsov估计理论被Bouehut
与Perthame[5}推广到用非线形退化守恒的抛物问题通近双曲守恒律(0.1.1)，并得到两
者嫡解的误差的显式表示，1 999年Coekburn与Gripenbeg[12]把双变t技巧得到非线形
退化守恒的抛物问题半群解在Ll惫义下如何显式依赖翰运流，扩散流及初始数据，在一
维时半群解就是嫡解，但对于多维情形，这种半群解是否仍为摘解还没有得到解决.
1985年DIPerna!31」引人多维单个双曲守恒律的测度值嫡解概念，用双变t技巧证
明了测度值摘解的唯一性与一个更深刻的结论:满足平均LI界，与初始数据相容的测度
值墒解在(x，t)的值就是嫡解在(x，t)值处的Dirac质t，称之为Dipema唯一性结果，
这也给出了多维单个双曲守恒律摘解与测度值摘解的联系.1983年r几rtar[s5}得到一
致Lco的估计序列的连续函数复合的弱星极限就是某一Young测度对该函数的作用(称
该y’oung测度为与估计序列相联系的(概率)测度族)并进一步说明了一致LOO序列有
子列几乎处处收敛到某函数与该序列的相应的Yoimg测度为Dirac质t等价.该结论与
DIPema唯一性结果给出了估计解序列收敛到多维单个双曲守恒律的摘解的一般框架:
10:构造多维单个双曲守恒律的估计格式，得到估计解序列，并证明它在Lco愈义
下为一致的.
20:与该估计解序列联系的Young测度满足DIPerna唯一性结论的条件.
30:由Diperna唯一性结果及Tartar结论，估计解序列有子列几乎处处收敛到多维
单个双曲守恒律的摘解.
一致L戈序列lu:}有子列(仍记为{丫}弱星收敛到某函数tL不能推得对任惫的光
滑函数f有f(丫)弱星收敛到f(司，这是由于紧性的缺乏引起，需要补偿一些紧性，称它
为补偿紧.1983年工成ar!85]得到补偿紧引理，并推得著名的Div一curl引理，他给出了
当一致L加估计序列的任意嫡消失测度在式
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 pr
oblem 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 defi
nition 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 kind
of 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
其次，在该区域附近通近不大好，如不能模拟物理上的展荡.所有这些都反映了解的正
则性的缺乏给数值研究带来困难.
Kruzkov【481在70年代用双变t技巧(Double variable teehnique)解决了多维单个双
曲守恒律(0 .1 .1)的摘解的适定性问题，即嫡解的存在性，唯一性及正规性结果，特别摘解
的Ll收缩性质.1976年Kuznetsov[sl]把双变t技巧用于双曲守恒律的估计问题，用抛
物正规(粘性方法)或用R‘上的矩形网格构造差分格式，并得到了相应的L‘收敛速度估
计.他的估计方法常称为Kuznetsov估计理论.1995年Kuznetsov估计理论被Bouehut
与Perthame[5}推广到用非线形退化守恒的抛物问题通近双曲守恒律(0.1.1)，并得到两
者嫡解的误差的显式表示，1 999年Coekburn与Gripenbeg[12]把双变t技巧得到非线形
退化守恒的抛物问题半群解在Ll惫义下如何显式依赖翰运流，扩散流及初始数据，在一
维时半群解就是嫡解，但对于多维情形，这种半群解是否仍为摘解还没有得到解决.
1985年DIPerna!31」引人多维单个双曲守恒律的测度值嫡解概念，用双变t技巧证
明了测度值摘解的唯一性与一个更深刻的结论:满足平均LI界，与初始数据相容的测度
值墒解在(x，t)的值就是嫡解在(x，t)值处的Dirac质t，称之为Dipema唯一性结果，
这也给出了多维单个双曲守恒律摘解与测度值摘解的联系.1983年r几rtar[s5}得到一
致Lco的估计序列的连续函数复合的弱星极限就是某一Young测度对该函数的作用(称
该y’oung测度为与估计序列相联系的(概率)测度族)并进一步说明了一致LOO序列有
子列几乎处处收敛到某函数与该序列的相应的Yoimg测度为Dirac质t等价.该结论与
DIPema唯一性结果给出了估计解序列收敛到多维单个双曲守恒律的摘解的一般框架:
10:构造多维单个双曲守恒律的估计格式，得到估计解序列，并证明它在Lco愈义
下为一致的.
20:与该估计解序列联系的Young测度满足DIPerna唯一性结论的条件.
30:由Diperna唯一性结果及Tartar结论，估计解序列有子列几乎处处收敛到多维
单个双曲守恒律的摘解.
一致L戈序列lu:}有子列(仍记为{丫}弱星收敛到某函数tL不能推得对任惫的光
滑函数f有f(丫)弱星收敛到f(司，这是由于紧性的缺乏引起，需要补偿一些紧性，称它
为补偿紧.1983年工成ar!85]得到补偿紧引理，并推得著名的Div一curl引理，他给出了
当一致L加估计序列的任意嫡消失测度在式
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 pr
oblem 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 defi
nition 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 kind
of 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
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