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Numerical boundary treatment for shock propagation in the fractional KdV-Burgers equation

Abstract

In this paper, we propose a numerical boundary treatment for simulating shock propagation in the fractional KdV-Burgers equation, based on a machine learning strategy. Numerical boundary treatment is particularly challenging due to the nonlinear and global interaction among the fractional diffusion, convection and dispersion. We select a suitable number of boundary points and interior points, and correct the numerical value at the boundary ones by linear combination of those at the interior ones. The large set of parameters are trained from a numerical reference shock solution connecting the end states 1 and 0, by ridge regression. Numerical tests demonstrate that the boundary treatment is effective in reflection suppression, and preserves the correct shock speed even under perturbed initial profile. Furthermore, using the above parameters, we construct a modified boundary treatment to simulate a range of different end states, with effectiveness checked numerically.

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Acknowledgements

This research is partially supported by NSFC under grant Nos. 11988102, 11832001, 11890681 and 11521202.

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Correspondence to Shaoqiang Tang.

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Appendix: parameters for \(K=10,L=20\)

Appendix: parameters for \(K=10,L=20\)

\(a_{jl}\) for \(j=1,l=N-19,\cdots,N\)

$$\begin{aligned}\begin{array}{rrrr} 5.96709588483390807\times 10^{-6}&{}-1.06056019213068138\times 10^{-6}&{}-2.65918837064829683\times 10^{-6}&{}-1.37970251223188428\times 10^{-6}\\ 3.28727863317240821\times 10^{-7}&{}8.41882484950767725\times 10^{-7}&{}-1.97963669303971328\times 10^{-7}&{}-1.95157732204002335\times 10^{-6}\\ -2.98585916582471497\times 10^{-6}&{}-2.17765705975681134\times 10^{-6}&{}4.97081058748227376\times 10^{-7}&{}3.70930549598096021\times 10^{-6}\\ 5.27563134303204731\times 10^{-6}&{}3.28740022779452256\times 10^{-6}&{}-2.58028144553970993\times 10^{-6}&{}-1.01838534310489384\times 10^{-5}\\ -1.49419955379447277\times 10^{-5}&{}-1.0919739306273805\times 10^{-5}&{}7.34403887061982576\times 10^{-6}&{}4.29200879683726264\times 10^{-5} \end{array}\end{aligned}$$

for \(j=2,l=N-19,\cdots,N\)

$$\begin{aligned}\begin{array}{rrrr} -1.80376900452201798\times 10^{-5}&{}1.91817480598611755\times 10^{-6}&{}5.92682456936428539\times 10^{-6}&{}3.13333810964239804\times 10^{-6}\\ 7.89147456481315122\times 10^{-7}&{}2.14582476710114549\times 10^{-6}&{}6.0572228384485059\times 10^{-6}&{}8.60190347921884895\times 10^{-6}\\ 6.07018220959293754\times 10^{-6}&{}-2.11092171849636617\times 10^{-6}&{}-1.20918120076063435\times 10^{-5}&{}-1.68991458882061637\times 10^{-5}\\ -1.00722209661598503\times 10^{-5}&{}9.83232467813916309\times 10^{-6}&{}3.59843702583298211\times 10^{-5}&{}5.3087101469844356\times 10^{-5}\\ 4.08683755902069688\times 10^{-5}&{}-1.97091198197665088\times 10^{-5}&{}-0.000139754026111402364&{}-0.00031786081515013402 \end{array}\end{aligned}$$

for \(j=3,l=N-19,\cdots,N\)

$$\begin{aligned} \begin{array}{rrrr} 8.65262198377192003\times 10^{-5}&{}-9.37704208105722453\times 10^{-6}&{}-3.11235330029904919\times 10^{-5}&{}-1.8674331272381723\times 10^{-5}\\ -4.81476068859341239\times 10^{-6}&{}-6.03000718516130725\times 10^{-6}&{}-2.01293789449023431\times 10^{-5}&{}-3.21268887972983366\times 10^{-5}\\ -2.60615629495024301\times 10^{-5}&{}2.75395674352157887\times 10^{-6}&{}4.14383009240148922\times 10^{-5}&{}6.33845042534324275\times 10^{-5}\\ 4.1872983249024011\times 10^{-5}&{}-3.18504605880706552\times 10^{-5}&{}-0.000134286273757923321&{}-0.000206780775969102635\\ -0.000167342354154886717&{}6.56635019496818633\times 10^{-5}&{}0.000546133817517070502&{}0.00127945470916221568\end{array}\end{aligned}$$

for \(j=4,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 3.36127225758924181\times 10^{-6}&{}-5.76315827808851304\times 10^{-6}&{}-9.53350033620676648\times 10^{-6}&{}-4.27948792444051652\times 10^{-6}\\ 7.63675291752229373\times 10^{-6}&{}1.87369909017633472\times 10^{-5}&{}2.0393826001737565\times 10^{-5}&{}8.17149742323896843\times 10^{-6}\\ -1.42546721437673334\times 10^{-5}&{}-3.515454096552175\times 10^{-5}&{}-3.9625612055790728\times 10^{-5}&{}-1.76806834247777211\times 10^{-5}\\ 2.80994471460814054\times 10^{-5}&{}7.94131860490665948\times 10^{-5}&{}0.000106046765910816412&{}7.60556366728713275\times 10^{-5}\\ -3.13825201215952196\times 10^{-5}&{}-0.000215788419379376727&{}-0.000452493767879921804&{}-0.000698779144773164444\end{array}\end{aligned}$$

for \(j=5,l=1,\cdots ,20\)

$$\begin{aligned}\begin{array}{rrrr} 2.73088394568609033\times 10^{-5}&{}-2.08912673876109197\times 10^{-6}&{}-1.5195077212687083\times 10^{-5}&{}-1.50658274606455898\times 10^{-5}\\ -6.24118885361973395\times 10^{-6}&{}5.29105764882720617\times 10^{-6}&{}1.31943184247486432\times 10^{-5}&{}1.28348872088137089\times 10^{-5}\\ 3.42943532913890269\times 10^{-6}&{}-1.09845897784511965\times 10^{-5}&{}-2.27510339326222518\times 10^{-5}&{}-2.39837858693117853\times 10^{-5}\\ -1.08946112327103243\times 10^{-5}&{}1.30476208808648424\times 10^{-5}&{}3.71101697641478369\times 10^{-5}&{}4.71930907810427786\times 10^{-5}\\ 3.25123373147508346\times 10^{-5}&{}-7.65954993048029623\times 10^{-6}&{}-6.04078406317308845\times 10^{-5}&{}-0.000100930542622343553\end{array}\end{aligned}$$

for \(j=6,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.00110659885425163349&{}-0.000135984903121426104&{}-0.000421454444203517753&{}-0.000247402019412601871\\ -3.75081078133153846\times 10^{-5}&{}-2.26085556628442565\times 10^{-5}&{}-0.000198950990925011193&{}-0.00038777046428971004\\ -0.000374742576206117662&{}-6.63778798376658499\times 10^{-5}&{}0.000415633133094018604&{}0.000760214020976359805\\ 0.000618618797794096226&{}-0.0001739509966868552&{}-0.00140512563080526795&{}-0.0024192151622396906\\ -0.00223119830174246369&{}0.000200106204433712144&{}0.00563110227646495664&{}0.0142511470436775013\end{array}\end{aligned}$$

for \(j=7,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} -0.000122645854754728316&{}-4.91550859555727168\times 10^{-5}&{}-7.26243100979984417\times 10^{-5}&{}-4.42822863271832026\times 10^{-5}\\ 8.68999288192161154\times 10^{-5}&{}0.000259395370391878998&{}0.000343571401193876387&{}0.000229126077607190654\\ -8.74194192299205787\times 10^{-5}&{}-0.000465562111804959644&{}-0.000665461505701619581&{}-0.000466795388885013864\\ 0.000186935083876456316&{}0.0010892228808074124&{}0.00177575576750496947&{}0.00165084632778594712\\ 0.000196537534328746578&{}-0.00281648796548455982&{}-0.00720504544498153446&{}-0.0123935494031932689\end{array}\end{aligned}$$

for \(j=8,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.00361768942286483982&{}-0.000413800285132311292&{}-0.00141206211080677994&{}-0.000907067550796972384\\ -0.000209205900393783065&{}-5.71605755828510903\times 10^{-5}&{}-0.000491641788087776479&{}-0.00102339661880351883\\ -0.00103394924954018923&{}-0.000229629356509176692&{}0.00107684288478016423&{}0.00202544298593906966\\ 0.00163045773251483382&{}-0.000588456236769275523&{}-0.0040416886678957464&{}-0.00685474337078876075\\ -0.00616392907328260509&{}0.00112213212579818163&{}0.01734194110367877&{}0.0432367057038350461\end{array}\end{aligned}$$

for \(j=9,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.0129374905953716351&{}-0.00178454515045794906&{}-0.00526923314052492493&{}-0.00308089269859637823\\ -0.000177207567478597017&{}0.000484466395763375108&{}-0.0014189042035158098&{}-0.00404162461896571869\\ -0.00480067885819748796&{}-0.00215856719696598314&{}0.00310646844496958573&{}0.0078677610959308978\\ 0.0080667230442055364&{}0.00110553305099504121&{}-0.0118014183082432948&{}-0.0244621389855264083\\ -0.0264809918133086311&{}-0.00595182451503434209&{}0.0467792683156444539&{}0.135646630796108558\end{array}\end{aligned}$$

for \(j=10,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.00605920662770449963&{}-0.0014290436792360325&{}-0.00385215416303510494&{}-0.00251439750122628425\\ 0.000536464441746236113&{}0.00303121894025295191&{}0.00327053334187016147&{}0.000855596005204854043\\ -0.00304020466063740233&{}-0.00611718407928972356&{}-0.00605184706570204966&{}-0.00183298407669103152\\ 0.00530621458693354177&{}0.0119208874576802668&{}0.0135339772911591549&{}0.00661116405315434648\\ -0.00945901585780703491&{}-0.0312852093069493492&{}-0.0518997335446071337&{}-0.0626912191231108473 \end{array} \end{aligned}$$

\(b_{jl}\) for \(j=1,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 1.15856179362907974\times 10^{-6}&{}-1.37055915129585336\times 10^{-7}&{}-7.48864536992893967\times 10^{-7}&{}-8.85454174733409756\times 10^{-7}\\ -7.27976134789503941\times 10^{-7}&{}-4.26901966105245514\times 10^{-7}&{}-1.00060280418828696\times 10^{-7}&{}1.68897307399201237\times 10^{-7}\\ 3.30150382999484145\times 10^{-7}&{}3.6449989227681089\times 10^{-7}&{}2.78644929892995812\times 10^{-7}&{}1.0086891557603018\times 10^{-7}\\ -1.2269922985226363\times 10^{-7}&{}-3.32361284443277268\times 10^{-7}&{}-4.60065788930993696\times 10^{-7}&{}-4.34931840456072392\times 10^{-7}\\ -1.88735426836924839\times 10^{-7}&{}3.38272797991024954\times 10^{-7}&{}1.19127343002557457\times 10^{-6}&{}2.3956839962067012\times 10^{-6}\end{array}\end{aligned}$$

for \(j=2,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} -4.30329911635055784\times 10^{-6}&{}3.48764355800493281\times 10^{-7}&{}2.59434225423932921\times 10^{-6}&{}3.1593603323632563\times 10^{-6}\\ 2.67275489532901091\times 10^{-6}&{}1.65636015349276408\times 10^{-6}&{}5.16588668403091155\times 10^{-7}&{}-4.59280697591064786\times 10^{-7}\\ -1.09715989814168771\times 10^{-6}&{}-1.32254314508777269\times 10^{-6}&{}-1.14630436536985312\times 10^{-6}&{}-6.52967209688487322\times 10^{-7}\\ 1.17621985920265002e-08&{}6.57351666454455148\times 10^{-7}&{}1.06754911832030046\times 10^{-6}&{}1.01986283143371255\times 10^{-6}\\ 3.05405017538621378\times 10^{-7}&{}-1.25042190847155212\times 10^{-6}&{}-3.76891375366169012\times 10^{-6}&{}-7.30519029098765491\times 10^{-6}\end{array}\end{aligned}$$

for \(j=3,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 1.94772201700530367\times 10^{-5}&{}-1.28579828497107601\times 10^{-6}&{}-1.14709740667233921\times 10^{-5}&{}-1.4248862461797729\times 10^{-5}\\ -1.23560393677861526\times 10^{-5}&{}-8.05238069058121306\times 10^{-6}&{}-3.09309122478503015\times 10^{-6}&{}1.2784386483929277\times 10^{-6}\\ 4.30093895845656077\times 10^{-6}&{}5.63466395143048397\times 10^{-6}&{}5.29504103737306361\times 10^{-6}&{}3.59462218271907979\times 10^{-6}\\ 1.08512053081620321\times 10^{-6}&{}-1.51161954747836835\times 10^{-6}&{}-3.38152944752790305\times 10^{-6}&{}-3.6972579645874014\times 10^{-6}\\ -1.6969692412591577\times 10^{-6}&{}3.23607395855496159\times 10^{-6}&{}1.15015134782122206\times 10^{-5}&{}2.32329211196849117\times 10^{-5}\end{array}\end{aligned}$$

for \(j=4,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} -2.22506771181927225\times 10^{-6}&{}-5.11419466162527601\times 10^{-7}&{}5.30256784494104565\times 10^{-7}&{}1.08480254242684371\times 10^{-6}\\ 1.2955948428942335\times 10^{-6}&{}1.26855242124374803\times 10^{-6}&{}1.06867158287946845\times 10^{-6}&{}7.29839668683986082\times 10^{-7}\\ 2.77912050620251208\times 10^{-7}&{}-2.53201147662468911\times 10^{-7}&{}-8.22208555766248928\times 10^{-7}&{}-1.38563499569198602\times 10^{-6}\\ -1.894006982944765\times 10^{-6}&{}-2.2815039317256282\times 10^{-6}&{}-2.45828494051470194\times 10^{-6}&{}-2.30627152859786243\times 10^{-6}\\ -1.67421489229736343\times 10^{-6}&{}-3.75179932952021907\times 10^{-7}&{}1.804775661814459\times 10^{-6}&{}5.08511577002261753\times 10^{-6}\end{array}\end{aligned}$$

for \(j=5,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 3.81418559557666998\times 10^{-6}&{}6.39093976245808866\times 10^{-7}&{}-1.42701214146135571\times 10^{-6}&{}-2.63409022939503831\times 10^{-6}\\ -3.17450917882877547\times 10^{-6}&{}-3.18282845322717003\times 10^{-6}&{}-2.75489151217868943\times 10^{-6}&{}-1.97321249074018546\times 10^{-6}\\ -9.21568109628360945\times 10^{-7}&{}3.08029187816517369\times 10^{-7}&{}1.6022101777068499\times 10^{-6}&{}2.80645936736068297\times 10^{-6}\\ 3.71210843204319369\times 10^{-6}&{}4.05787123926464708\times 10^{-6}&{}3.53990731455883702\times 10^{-6}&{}1.82340651435592229\times 10^{-6}\\ -1.44191627746274509\times 10^{-6}&{}-6.5982729649225584\times 10^{-6}&{}-1.39534318523984799\times 10^{-5}&{}-2.37621951432726929\times 10^{-5}\end{array}\end{aligned}$$

for \(j=6,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.000240974324586248938&{}-1.81573013546840852\times 10^{-5}&{}-0.000144555718297340432&{}-0.000178051718245355911\\ -0.000153089322551191754&{}-9.81665048931708865\times 10^{-5}&{}-3.54811892289556805\times 10^{-5}&{}1.91983044700044842\times 10^{-5}\\ 5.62303811128859351\times 10^{-5}&{}7.13810737124962033\times 10^{-5}&{}6.49829942791170425\times 10^{-5}&{}4.118752405490491\times 10^{-5}\\ 7.24231151688054406\times 10^{-6}&{}-2.73572077097092134\times 10^{-5}&{}-5.18403682876483689\times 10^{-5}&{}-5.51587232361751479\times 10^{-5}\\ -2.69815211497760824\times 10^{-5}&{}4.12886875179245037\times 10^{-5}&{}0.000155562256602214939&{}0.000318364920194253711\end{array}\end{aligned}$$

for \(j=7,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} -7.05388046999852025\times 10^{-5}&{}-7.42321773221023896\times 10^{-7}&{}3.44116056536350183\times 10^{-5}&{}4.53747668333819796\times 10^{-5}\\ 4.10305222923393811\times 10^{-5}&{}2.86252442481660968\times 10^{-5}&{}1.35936247662989933\times 10^{-5}&{}-4.15568843960512856\times 10^{-7}\\ -1.12163650232785126\times 10^{-5}&{}-1.77359790761738086\times 10^{-5}&{}-1.98701330553923548\times 10^{-5}&{}-1.84376977676922213\times 10^{-5}\\ -1.50075689745697876\times 10^{-5}&{}-1.157246373667613\times 10^{-5}&{}-1.02050984870293531\times 10^{-5}&{}-1.27497057820708171\times 10^{-5}\\ -2.04948121903767859\times 10^{-5}&{}-3.38499343103427369\times 10^{-5}&{}-5.21716338482884642\times 10^{-5}&{}-7.38342361597292215\times 10^{-5}\end{array}\end{aligned}$$

for \(j=8,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.000770689704498828598&{}-4.5063750833779108\times 10^{-5}&{}-0.000449794161224371632&{}-0.000565769408560374263\\ -0.000498401302486923842&{}-0.000334584323509505934&{}-0.000141857381380272639&{}3.16520391541149151\times 10^{-5}\\ 0.000156087826956657677&{}0.000217596875653393377&{}0.000215670534758508796&{}0.000160695472181201344\\ 7.16350546218094637\times 10^{-5}&{}-2.65090089852571419\times 10^{-5}&{}-0.000105495455921804913&{}-0.00013668950680429076\\ -9.38591440938442347\times 10^{-5}&{}4.40715876604672841\times 10^{-5}&{}0.000290518258262271187&{}0.000649436102881048929\end{array}\end{aligned}$$

for \(j=9,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.00269286272560756738&{}-0.000223826051374996618&{}-0.00164169988052013873&{}-0.00201056607147178388\\ -0.00171994906026385458&{}-0.0010925266015740225&{}-0.000380308845214852689&{}0.00023708370519939456\\ 0.000649768402938976134&{}0.000809973740798323365&{}0.000722344476939336096&{}0.000435095547424410416\\ 3.1830959613417515\times 10^{-5}&{}-0.000377747539720010655&{}-0.000668605355004438131&{}-0.000711552123765074747\\ -0.000384352364025264653&{}0.000416884071337631347&{}0.0017666041168752655&{}0.00370110435857568974\end{array}\end{aligned}$$

for \(j=10,l=1,\cdots ,20\)

$$\begin{aligned} \begin{array}{rrrr} 0.00073892456722970761&{}-9.03746881086875284\times 10^{-5}&{}-0.000501782487294801936&{}-0.000616791388570555829\\ -0.000541403032444684072&{}-0.000363637415986265658&{}-0.000154176276519580327&{}3.37431902308869971\times 10^{-5}\\ 0.00016573136549614709&{}0.000225824702309761736&{}0.000212780034852155401&{}0.000135518351461006022\\ 1.04260940384744983\times 10^{-5}&{}-0.000139781419626450757&{}-0.00028739894444162773&{}-0.000401567705584427825\\ -0.000449744522484811732&{}-0.000399033421264799329&{}-0.000218763712698090776&{}0.000115321963429921808 \end{array}\end{aligned}$$

\(r_j\) for \(j=1,\cdots ,10\)

$$\begin{aligned}\begin{array}{r} 5.4449024039008694\times 10^{-7}\\ 4.7036819024312042\times 10^{-7} \\ 7.84057381289423876\times 10^{-7} \\ 5.03258956600924976\times 10^{-7} \\ 5.74503313084172221\times 10^{-7} \\ 3.76184845780128559\times 10^{-6} \\ -2.96552867726106834\times 10^{-7} \\ 1.08940244047946874\times 10^{-5} \\ 3.71477635139735369\times 10^{-5} \\ 1.2128348593418037\times 10^{-5} \end{array}\end{aligned}$$

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Guan, X., Zhang, Q. & Tang, S. Numerical boundary treatment for shock propagation in the fractional KdV-Burgers equation. Comput Mech (2021). https://doi.org/10.1007/s00466-021-02089-z

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Keywords

  • Fractional KdV-Burgers equation
  • Shock wave
  • Boundary treatment
  • Machine learning
  • Ridge regression