The High Resolution Difference Schemes Based Onaveraged Inverse Mapping And Theirapplications

The main purpose of this paper is to research the numerical methods for nonlinearhyperbolic conservation laws which is easy to compute, uniformly high-order accuracy,high resolution and non-oscillatory. It is well-known. The main difficulty to solve theinitial value problem of hyperbolic conservation laws is that no matter how theinitial value function is smooth, its solution may include discontinuous. In theprocess of constructing finite difference schemes which to solve hyperbolicconservation laws, the calculation of first-order schemes may oversmear a strongdiscontinuity and that of the high-order schemes may produce numericaloscillatory at discontinuity. In order to avoid numerical oscillatory, the correctionmethod is usually used by the TVD limiter function, but the accuracy of the schemes atdiscontinuity and extreme points will decrease to first-order. Based on these, in thispaper,we present a kind of new method that may overcome shortage of accuracy atdiscontinuity, which the method is called averaged inverse mapping. Several kinds ofhigh-order accuracy and high-resolution difference schemes of solving hyperbolicconservation laws is constructed by applying to the central difference scheme, the fluxsplitting scheme and the scheme of undetermined coefficient method. The highresolution of these schemes are verified through a lot of numerical experiments. Themain achievements of work is as follows:1. Based on the averaged inverse mapping method instead of the TVD limiterfunction method to select the numerical derivatives, by taking the staggered andnon-staggered Lax-Friedrichs scheme as basic building block and using piecewise linearreconstruction instead of piecewise constant approximation, a class of second-orderstaggered and non-staggered central difference schemes is respectively constructed tosolve nonlinear hyperbolic conservation laws, which based on averaged inversemapping. Furthermore, by applying piecewise cubic polynomial reconstruction andcombining fourth-order Runge-Kutta NCE methods to calculate the point value at themiddle time, a class of fourth-order central difference schemes to solve nonlinearhyperbolic conservation laws is presented. Then, the extension to the system situation isimplemented by using component-wise manner. Finally, a lot of typical numericalexamples are given to verify the properties of simple form, uniformly high-order accuracy, high resolution and stability of the present schemes. Comparing thecalculation effects of numerical derivatives that based on averaged inverse mappingmethod and TVD limiter function method, the high resolution of the schemes in thisChapter is demonstrated.2. Based on averaged inverse mapping and flux splitting method, we construct thenumerical derivatives of flux in positive and negative parts. Then a class ofsecond-order and third-order flux splitting schemes to solve nonlinear hyperbolicconservation laws is respectively presented. The extension to the system situation isimplemented by using component-wise manner. Then, a lot of typical numericalexamples are given to verify the properties of simple form, uniformly high-orderaccuracy, high resolution and stability of these schemes. Finally, we compare andanalysis the results of calculation of numerical derivatives that based on averagedinverse mapping method and TVD limiter function method.3. The second-order and third-order difference approximation of the first-orderspace derivatives of flux function is respectively deduced by applying an undeterminedcoefficient method. Then, based on averaged inverse mapping method and TVD limiterfunction method to correct the difference approximation of derivatives, a kind ofsecond-order and third-order non-oscillatory difference schemes is obtained by usingRunge-Kutta TVD time discretization methods. A lot of typical numerical examples aregiven to show the properties of simple form, easy to calculate, high resolution andstability of the schemes in this chapter.4. Based on the closely relationship between shallow water wave equations andhyperbolic conservation laws, we extend the second-order staggered non-oscillatorycentral difference schemes which solving hyperbolic conservation laws in chapter2toshallow water wave equations. Then a second-order staggered non-oscillatory centraldifference schemes to solve shallow water wave equations without source terms isobtained. A standard numerical example is given to show the non-oscillatory propertyof this scheme.5. The third classes of schemes that based on averaged inverse mapping in thispaper are compared. And then, the further work is present.