WENO Schemes With Order-Preserving Mappings

This dissertation focuses on the development of weighted essentially non-oscillatory(WENO)schemes.Since WENO schemes can achieve high order accuracy in smooth regions while preventing spurious oscillations near discontinuities,they have become one of the most popular methods for approximating solutions to hyperbolic conservation laws.However,in long-run simulations,it seems to be very different and extremely challenging for WENO schemes to obtain considerably high resolutions and meanwhile avoid spurious oscillations.A number of studies have indicated that most of the existing WENO schemes suffer from either losing high resolutions or generating spurious oscillations on solving hyperbolic problems with discontinuities at large output times.During the last decade,this issue has attracted considerable scholarly interest but the way of completely addressing it has remained unclear up to now.This thesis conducts a systematical investigation on various existing mapped WENO schemes and reveals the essential reason of the issue mentioned above.It is actually caused by that the mapping functions of these schemes can not preserve the order of the nonlinear weights of the stencils.In other words,the nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils.Therefore,we develop the novel concept of order-preserving(OP)mapping,and this provides a new direction for both devising new WENO schemes and improving the existing ones.Extensive numerical experiments demonstrate that the WENO schemes which adopt OP mappings exhibit remarkable advantages for long-run simulations,such as attaining high resolutions and in the meantime avoiding spurious oscillations near discontinuities,as well as obtaining high resolutions on solving problems with high-order critical points.Furthermore,even in short-time simulations,they also gain several considerable enhancements,such as improving resolutions in the regions with high-frequency smooth waves,removing or significantly reducing post-shock numerical oscillations on solving the two-dimentional problems with shock waves,as well as greatly limiting numerical oscillations in the framework of component-wise reconstruction.The above work explores a brand new way for the study in the field of WENO schemes.It is be preceded by another two relevant studies on enhancing the resolution and robustness and improving the efficiency of the mapped WENO schemes.Firstly,we originally build an adaptive mapping function and hence devise a new family of mapped WENO schemes.One outstanding scheme with fine-tuned parameters is found.It illustrates a significant advantage on solving problems with discontinuities,that is,it produces numerical solutions with very high resolutions without generating spurious oscillations,especially for long-run simulations.Secondly,aiming at reducing the extra computational cost brought by existing mapping processes,we construct a novel approximate constant mapping function that meets the overall existing criteria for a proper mapping function.The resultant mapped WENO scheme brings a notable improvement of efficiency.Indeed,it is the first mapped WENO scheme that can decrease the extra computational cost compared to WENO-JS to less than 5%.Specifically,our main contributions are as follows.(1)Propose a series of mapped WENO schemes using adaptive mappings.By constructing an approximation of the signum function and some adaptive control functions,we devise a new family of mapping functions that introduces the adaptive nature and provides a wider selection of the parameters in the resultant schemes.Theoretical analysis and numerical results show that the new schemes attain optimal convergence rates of accuracy in smooth regions regardless of critical points.Particularly,one recommended scheme makes a significant breakthrough,that is,it achieves very high resolutions and removes spurious oscillations at the same time on solving problems with discontinuities at long output times.(2)Propose an efficient mapped WENO scheme using an approximate constant mapping.The primary reason for the computational cost increase of the existing mapped WENO schemes is that their specified complicated mapping procedures require a large number of extra mathematical operations.Thus,by employing the aforementioned approximation of the signum function again,we design an approximate constant mapping function.It satisfies the overall existing criteria for a proper mapping function while almost uses only one assignment operation.Thus,the resultant scheme can obtain optimal convergence orders even at critical points and it is very efficient.Extensive numerical tests show that its extra computational cost compared to WENO-JS is less than 5%and this is far lower than other mapped WENO schemes.(3)Build the original concept of order-preserving(OP)mapping and construct an OPMapped WENO scheme accordingly.Through a systematic analysis,we first find that the order of the nonlinear weights has been changed in all existing mapping processes.This means that these mappings can probably increase weights of non-smooth substencils and decrease weights of smooth substencils.Obviously,this is positively harmful to gaining high-resolution and robustness for the WENO schemes,especially in long-run simulations.Therefore,we innovatively give the definition of the OP mapping and hence propose a new mapped WENO scheme.It achieves the optimal convergence order of accuracy even at critical points,and most importantly,it has significantly high resolution but does not generate spurious oscillation near discontinuities even if the output time is large.(4)Propose a general method to introduce OP mappings to existing mapped WENO schemes.We create a generalized implementation of improving existing mappings by extending the OP criterion.Theoretical analysis and numerical experiments have been performed to prove that the improved mapped WENO schemes can get the same convergence orders as the existing ones.In long-run simulations,they can amend the drawback of generating spurious oscillations or failing to obtain high resolutions on solving problems with discontinuities.They can also address the issue of failing to obtain high resolutions on solving problems with high-order critical points.In addition,they can remove or significantly reduce the post-shock numerical oscillations in simulating the two-dimensional problems with strong shock waves.And also,they can greatly limit the numerical oscillations for the cases of employing the component-wise reconstruction method.(5)Build the concept of locally-order-preserving(LOP)mappings and propose a general method to construct mapped WENO schemes with LOP mappings accordingly.We find that the requirement to ensure that the mapping is OP on the whole interval is unnecessary and even too strict,and it results in the low resolutions in the regions of high-frequency smooth waves and limits the availability of the idea of OP to other versions of WENO schemes.Thus,we further propose the definition of LOP mapping,that is,the OP is only required in the reconstruction of current location.Then,by using a new proposed posteriori adaptive technique,we also propose a generalized implementation of applying this LOP property to obtain the improved mappings from existing ones.The corresponding new schemes gain all advantages of the improved schemes using OP mappings mentioned earlier.In addition,they can improve resolutions in the regions with high-frequency smooth waves.(6)Achieve general improvements in other versions of WENO schemes.To date,far too little attention has been paid to the long-run simulations of other versions of WENO schemes.Indeed,they also terribly suffer from either losing high resolutions or generating numerical oscillations.To fix this issue,inspired by extensive numerical observations,we firstly rewrite the formulae of various nonlinear weights in a uniform form from a mapping perspective and propose the concept of generalized mapping.Then,by extending the former two generalized methods,we propose two groups of improved generalized mapped WENO schemes.They have exactly the same merits as those of the corresponding improved traditional mapped WENO schemes,respectively.