A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

Leilei Wei; Yinnian He

doi:10.3934/dcdsb.2020319

Article Contents

2021, Volume 26, Issue 9: 4907-4926. Doi: 10.3934/dcdsb.2020319

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A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

Leilei Wei^1, , and
Yinnian He^2,

1.
College of Science, Henan University of Technology, Zhengzhou 450001, China
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

^* Corresponding author: Leilei Wei
^* Corresponding author: Leilei Wei

Received: March 2020

Revised: August 2020

Early access: November 2020

Published: September 2021

Supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology(2018RCJH10), the Training Plan of Young Backbone Teachers in Henan University of Technology(21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province (2019GGJS094), the Innovative Funds Plan of Henan University of Technology, Foundation of Henan Educational Committee(19A110005) and the National Natural Science Foundation of China (11771348, 11861068)

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Abstract

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order $ O(h^{k+1}+\tau^{2-\alpha}) $, where $ h, \tau $ and $ k $ are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

Keywords:

Tempered fractional diffusion equations,
local discontinuous Galerkin method,
stability,
error estimates.

Mathematics Subject Classification: Primary: 65M12, 65M06; Secondary: 35S10.

Citation:

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Figure 1. The evolution of the solution for $ \alpha = 0.3 $. $ \tau = 0.001, h = 0.01, k = 2. $

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Figure 2. The evolution of the solution for $ \alpha = 0.8 $. $ \tau = 0.001, h = 0.01, k = 2. $

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Table 1. Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $ \delta = 0.3, \gamma = 2, M = 10^3, T = 1 $

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.3$	$\alpha=0.1$	$P^0$	5	3.600997655347402E-002	-	8.470765811325168E-002	-
			10	1.751495701129877E-002	1.04	4.247840062543977E-002	1.00
			20	8.698273697461307E-003	1.01	2.125369555415509E-002	1.00
			40	4.341798450317296E-003	1.00	1.062863299729018E-002	1.00
		$P^1$	5	9.943585411573410E-003	-	2.757075846380118E-002	-
			10	3.566273983250412E-003	1.48	1.080003816776576E-002	1.35
			20	1.086919816367809E-003	1.71	3.420222759716512E-003	1.66
			40	2.902709222939700E-004	1.90	9.193407604523585E-004	1.90
		$P^2$	5	7.990640423350242E-004	-	3.417434847568336E-003	-
			10	8.626266947702263E-005	3.21	3.694158436061931E-004	3.20
			20	1.044514634698983E-005	3.04	4.454609652087860E-005	3.05
			40	1.296172202016491E-006	3.01	5.508580340890919E-006	3.01
	$\alpha=0.6$	$P^0$	5	3.594827779072097E-002	-	8.455751510071857E-002	-
			10	1.750788942725992E-002	1.04	4.246067080980019E-002	1.00
			20	8.697412259699867E-003	1.01	2.125151852856288E-002	1.00
			40	4.341693080827747E-003	1.00	1.062836627483681E-002	1.00
		$P^1$	5	9.921671652323443E-003	-	2.759417252018699E-002	-
			10	3.563738012224470E-003	1.48	1.080391242567882E-002	1.35
			20	1.086707146130006E-003	1.71	3.420472591014495E-003	1.66
			40	2.902558645272467E-004	1.90	9.194266256405403E-004	1.90
		$P^2$	5	7.988664990181788E-004	-	3.416510378879402E-003	-
			10	8.626077461730029E-005	3.21	3.694001433403462E-004	3.20
			20	1.044545573669924E-005	3.04	4.454547869921856E-005	3.05
			40	1.298706242251523E-006	3.01	5.508452148315928E-006	3.01
	$\alpha=0.8$	$P^0$	5	3.592329974624144E-002	-	8.449616965675322E-002	-
			10	1.750510512702476E-002	1.04	4.245362255057922E-002	1.00
			20	8.697106637946338E-003	1.01	2.125073937587364E-002	1.00
			40	4.341672465121832E-003	1.00	1.062831369241851E-002	1.00
		$P^1$	5	9.912664929559693E-003	-	2.760460772303402E-002	-
			10	3.562659912073956E-003	1.48	1.080630043834668E-002	1.35
			20	1.086605203698557E-003	1.71	3.421287359547415E-003	1.66
			40	2.902460763410707E-004	1.90	9.201811924158254E-004	1.90
		$P^2$	5	7.988141577793738E-004	-	3.416151504204671E-003	-
			10	8.626566626691957E-005	3.21	3.693913312302985E-004	3.21
			20	1.046396713324052E-005	3.04	4.454319799217823E-005	3.05
			40	1.436914611732205E-006	2.86	5.619072907782352E-006	2.99

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Table 2. Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\delta = 0.1$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.1$	$\alpha=0.1$	$P^0$	5	3.600997655347402E-002	-	8.470765811325168E-002	-
			10	1.751495701129877E-002	1.04	4.247840062543977E-002	1.00
			20	8.698273697461307E-003	1.01	2.125369555415509E-002	1.00
			40	4.341798450317296E-003	1.00	1.062863299729018E-002	1.00
		$P^1$	5	9.125860752324633E-003	-	3.386112146946083E-002	-
			10	2.295048237777114E-003	1.99	8.756550512034528E-003	1.95
			20	5.745423502445809E-004	2.00	2.207822490277067E-003	1.98
			40	1.436832777579926E-004	2.00	5.554382129445423E-004	1.99
		$P^2$	5	9.050669939275690E-004	-	4.298666679244556E-003	-
			10	1.151488108051345E-004	2.97	5.375579180586712E-004	3.00
			20	1.445721437410244E-005	2.99	6.924567827167210E-005	2.96
			40	1.809466716341844E-006	3.00	8.720509756160153E-006	2.99
	$\alpha=0.6$	$P^0$	5	3.594827779072097E-002	-	8.455751510071857E-002	-
			10	1.750788942725992E-002	1.04	4.246067080980019E-002	1.00
			20	8.697412259699867E-003	1.01	2.125151852856288E-002	1.00
			40	4.341693080827747E-003	1.00	1.062836627483681E-002	1.00
		$P^1$	5	9.121793113875693E-003	-	3.384039722034468E-002	-
			10	2.294831526335838E-003	1.99	8.755484555290433E-003	1.95
			20	5.745294690001247E-004	2.00	2.207828785762922E-003	1.98
			40	1.436825323167711E-004	2.00	5.555100933167800E-004	1.99
		$P^2$	5	9.047609877074052E-004	-	4.297245174444950E-003	-
			10	1.151394966919664E-004	2.97	5.375058641304305E-004	3.00
			20	1.445715309857497E-005	2.99	6.924406859557769E-005	2.96
			40	1.811270951788605E-006	3.00	8.720463021409203E-006	2.99
	$\alpha=0.8$	$P^0$	5	3.592329974624144E-002	-	8.449616965675322E-002	-
			10	1.750510512702476E-002	1.04	4.245362255057922E-002	1.00
			20	8.697106637946338E-003	1.01	2.125073937587364E-002	1.00
			40	4.341672465121832E-003	1.00	1.062831369241851E-002	1.00
		$P^1$	5	9.120195825181981E-003	-	3.383289362730243E-002	-
			10	2.294751340921533E-003	1.99	8.755789596454649E-003	1.95
			20	5.745261460924934E-004	2.00	2.208566384755528E-003	1.98
			40	1.436839091653192E-004	2.00	5.562652869576801E-004	1.99
		$P^2$	5	9.046409059470311E-004	-	4.297126983902264E-003	-
			10	1.151376194022218E-004	2.97	5.374880423776833E-004	3.00
			20	1.447012828175748E-005	2.99	6.924379906001369E-005	2.96
			40	1.912757116440075E-006	2.99	8.835769827388040E-006	2.99

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Table 3. Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\delta = 0.3$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.3$			$\alpha=0.1$	$P^0$	5	6.559598407976026E-002	-	0.141942066666751	-
					10	2.141331033721447E-002	1.61	5.734028406382948E-002	1.30
					20	9.793316970413680E-003	1.12	2.815183989101257E-002	1.02
		40			4.809485138977877E-003	1.02	1.407535025139766E-002	1.00
		$P^1$		5	9.334237440748634E-003	-	3.145978397121429E-002	-
				10	3.638112329117496E-003	1.36	1.339194646885009E-002	1.23
				20	1.106145242636678E-003	1.72	4.300814221832788E-003	1.64
				40	2.952217379902808E-004	1.91	1.195193749833484E-003	1.85
		$P^2$		5	1.010581921645606E-003	-	5.126434823465115E-003	-
				10	1.068201999084790E-004	3.24	5.971277580741609E-004	3.10
				20	1.389860346772117E-005	2.94	8.309553534868077E-005	2.85
	40			1.770704698351656E-006	2.97	1.076235396982232E-005	2.95
	$\alpha=0.6$	$P^0$	5	6.519959540296516E-002	-	0.141364976011094	-
			10	2.137597683554834E-002	1.60	5.728856602677920E-002	1.31
			20	9.788419793457237E-003	1.12	2.814456788448857E-002	1.02
			40	4.808850888477931E-003	1.01	1.407438548005000E-002	1.00
		$P^1$	5	9.324055653453555E-003	-	3.148494472151778E-002	-
			10	3.637084108871119E-003	1.36	1.339218081832141E-002	1.23
			20	1.106059577072876E-003	1.72	4.300878545495546E-003	1.64
			40	2.952158819276407E-004	1.91	1.195178864173474E-003	1.85
		$P^2$	5	1.010440003914728E-003	-	5.125652210367676E-003	-
			10	1.068162274307204E-004	3.23	5.971411497308199E-004	3.10
			20	1.389837000772890E-005	2.94	8.309427296208810E-005	2.85
			40	1.770454293076160E-006	2.97	1.076230665963618E-005	2.95
	$\alpha=0.8$	$P^0$	5	6.458019587494233E-002	-	0.140458685661883	-
			10	2.131839260620296E-002	1.60	5.720848587853570E-002	1.31
			20	9.780967395542408E-003	1.12	2.813343523512223E-002	1.01
			40	4.807925152620669E-003	1.02	1.407295053444370E-002	1.00
		$P^1$	5	9.308221701165681E-003	-	3.152477270193341E-002	-
			10	3.635461850529788E-003	1.36	1.339294605022746E-002	1.24
			20	1.105916081128332E-003	1.72	4.301517739581207E-003	1.64
			40	2.952039890812090E-004	1.91	1.195672409974230E-003	1.85
		$P^2$	5	1.010240608178436E-003	-	5.124773588667655E-003	-
			10	1.068123075153382E-004	3.23	5.971640034306184E-004	3.10
			20	1.390323465425386E-005	2.94	8.309265950586455E-005	2.85
			40	1.809513967454312E-006	2.98	1.076227541254043E-005	2.95

| Show Table

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Table 4. Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\delta = 0.1$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.1$			$\alpha=0.1$	$P^0$	5	4.554426032174388E-002	-	0.115718177138195	-
					10	1.907614555969082E-002	1.25	5.320421957088832E-002	1.12
					20	9.138240767269679E-003	1.06	2.598259449662434E-002	1.03
		40			4.518863802087745E-003	1.01	1.291377789093143E-002	1.00
		$P^1$		5	8.849830180737654E-003	-	3.969655576082662E-002	-
				10	2.670888776568979E-003	1.73	1.117674633593155E-002	1.83
				20	6.856209305073176E-004	1.96	2.993783590210047E-003	1.90
				40	1.726263868565285E-004	1.99	7.624154117570892E-004	1.97
		$P^2$		5	1.082053891229540E-003	-	5.632531555489165E-003	-
				10	1.177827066011148E-004	3.20	7.864352387506149E-004	2.84
				20	1.464825722091804E-005	3.01	9.550948635494300E-005	3.04
	40			1.829209283100811E-006	3.00	1.207411590455412E-005	2.98
	$\alpha=0.6$	$P^0$	5	4.543056490078846E-002	-	0.115474889448670	-
			10	1.906295169538061E-002	1.25	5.318082033209643E-002	1.12
			20	9.136635333555037E-003	1.06	2.597982073437711E-002	1.03
			40	4.518663596834055E-003	1.01	1.291343451385024E-002	1.00
		$P^1$	5	8.846862605510398E-003	-	3.970373991829590E-002	-
			10	2.670694860720131E-003	1.73	1.117614841378017E-002	1.83
			20	6.856086070180152E-004	1.96	2.993777705012829E-003	1.90
			40	1.726256192263683E-004	1.99	7.623984725640964E-004	1.97
		$P^2$	5	1.081894553798753E-003	-	5.631794818602251E-003	-
			10	1.177789297367418E-004	3.20	7.864212743115771E-004	2.84
			20	1.464811122465675E-005	3.01	9.550366030672275E-005	3.04
			40	1.828970584365630E-006	3.00	1.207133596128895E-005	2.99
	$\alpha=0.8$	$P^0$	5	4.525430019654393E-002	-	0.115095429944861	-
			10	1.904269469568265E-002	1.25	5.314471217556128E-002	1.12
			20	9.134222951010450E-003	1.06	2.597562042326319E-002	1.03
			40	4.518387505390618E-003	1.01	1.291295179596993E-002	1.00
		$P^1$	5	8.842271383522345E-003	-	3.971560941693522E-002	-
			10	2.670392762456530E-003	1.73	1.117577463774522E-002	1.83
			20	6.855884088997946E-004	1.96	2.994298251659394E-003	1.90
			40	1.726244848834557E-004	1.99	7.628868021073432E-004	1.97
		$P^2$	5	1.081660399859714E-003	-	5.630991561418289E-003	-
			10	1.177745623909475E-004	3.20	7.864030096010258E-004	2.84
			20	1.465280637729603E-005	3.01	9.565506409087154E-005	3.04
			40	1.866808245534035E-006	3.00	1.214811102974098E-005	2.99

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Table 5. Temporal accuracy test using piecewise $P^2$ polynomials for the scheme (9) with generalized alternating numerical fluxes when $N = 100, T = 1.$

$\delta$	$\alpha$	$\tau$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.1$	$\alpha=0.5$	0.04	8.608763604447880E-006	-	1.219684581693636E-005	-
		0.02	3.086574970641430E-006	1.48	4.409647689024299E-006	1.47
		0.01	1.109311333958263E-006	1.48	1.667659212722938E-006	1.40
		0.005	4.122054755449973E-007	1.43	6.985320445991206E-007	1.26
	$\alpha=0.7$	0.04	2.391687146347450E-005	-	3.383563665176892E-005	-
		0.02	9.853827361471093E-006	1.28	1.395429884359922E-005	1.28
		0.01	4.116933701786636E-006	1.26	5.856072709420346E-006	1.26
		0.005	1.782609365838843E-006	1.21	2.587548893207003E-006	1.18
$\delta=0.3$	$\alpha=0.5$	0.04	8.608382726939050E-006	-	1.217526939467639E-005	-
		0.02	3.085511558119693E-006	1.48	4.377333736760303E-006	1.48
		0.01	1.106348076087462E-006	1.48	1.601411114160456E-006	1.45
		0.005	4.041621399151297E-007	1.45	6.654877294648420E-007	1.27
	$\alpha=0.7$	0.04	2.391673470774867E-005	-	3.382490687975359E-005	-
		0.02	9.853494675524166E-006	1.28	1.393619582018557E-005	1.28
		0.01	4.116136624036466E-006	1.26	5.832318129950220E-006	1.26
		0.005	1.780767052630476E-006	1.21	2.542037339958725E-006	1.20

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Table 6. Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.2$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.2$	$\alpha=0.3$	$P^0$	5	1.582340814827774E-003	-	3.587747814566711E-003	-
			10	6.072920572041250E-004	1.38	1.361992544606309E-003	1.39
			20	2.781722995839050E-004	1.12	6.593363827866452E-004	1.05
			40	1.358688098753038E-004	1.03	3.262244021583708E-004	1.01
		$P^1$	5	2.976903263874860E-004	-	8.888076108539667E-004	-
			10	9.380682880588326E-005	1.67	3.747599073017630E-004	1.25
			20	2.570892164256014E-005	1.87	1.197734627369952E-004	1.65
			40	6.616980217734994E-006	1.96	3.372607623089679E-005	1.83
		$P^2$	5	3.744225938986966E-005	-	2.066750403204734E-004	-
			10	4.242629022151605E-006	3.14	2.905747077063567E-005	2.83
			20	5.108240307324746E-007	3.05	3.733062412131737E-006	2.96
			40	1.404276778729993E-007	1.86	4.548930764244681E-007	3.03
	$\alpha=0.5$	$P^0$	5	1.576014035107718E-003	-	3.577323483172182E-003	-
			10	6.065883866688050E-004	1.38	1.361430273966585E-003	1.39
			20	2.780870829316098E-004	1.12	6.589782886991372E-004	1.05
			40	1.358586702912018E-004	1.03	3.262537733518698E-004	1.01
		$P^1$	5	2.975626899907474E-004	-	8.888043631958508E-004	-
			10	9.379482889619085E-005	1.68	3.746166818088215E-004	1.25
			20	2.570772139874646E-005	1.87	1.196172349767182E-004	1.65
			40	6.615791862540715E-006	1.96	3.356898366286719E-005	1.83
		$P^2$	5	3.743605733712328E-005	-	2.068108840347286E-004	-
			10	4.240769769090305E-006	3.14	2.921541320911186E-005	2.82
			20	4.961774292224783E-007	3.09	3.890639954380062E-006	2.90
			40	7.051223319124203E-008	2.81	5.312153408062071E-007	2.87
	$\alpha=0.7$	$P^0$	5	1.569573150535817E-003	-	3.566532563472458E-003	-
			10	6.058781928822529E-004	1.37	1.361570566765813E-003	1.39
			20	2.780022433132069E-004	1.12	6.584717897009694E-004	1.04
			40	1.358494586510882E-004	1.03	3.265674113640243E-004	1.01
		$P^1$	5	2.974342179950080E-004	-	8.886664664762281E-004	-
			10	9.378333026032094E-005	1.67	3.743331911596685E-004	1.25
			20	2.570912149705274E-005	1.87	1.193198577654686E-004	1.65
			40	6.624911739109129E-006	1.96	3.327059953717988E-005	1.84
		$P^2$	5	3.743175301428064E-005	-	2.070928579500273E-004	-
			10	4.255010460194694E-006	3.13	2.951598305808386E-005	2.81
			20	6.067267031675130E-007	2.81	4.189879914070192E-006	2.82
			40	8.591886309964037E-008	2.82	6.269178180106001E-007	2.74

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Table 7. Spatial accuracy test on uniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.6$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.6$	$\alpha=0.3$	$P^0$	5	2.263070303841627E-003	-	4.577360163679048E-003	-
			10	6.707774222704496E-004	1.75	1.436684775454216E-003	1.67
			20	2.853444715298516E-004	1.23	6.606848412090876E-004	1.12
			40	1.367407768126421E-004	1.06	3.273628226099778E-004	1.01
		$P^1$	5	2.974509312553489E-004	-	7.150301084008492E-004	-
			10	1.271245501318707E-004	1.22	3.981226597552988E-004	0.84
			20	5.095203810569959E-005	1.32	1.756826753133757E-004	1.18
			40	1.628155726102348E-005	1.65	6.113921628532708E-005	1.52
		$P^2$	5	4.741117314917393E-005	-	2.322474249148421E-004	-
			10	5.251072284968953E-006	3.17	3.482052702302946E-005	2.73
			20	5.487061550738519E-007	3.26	4.391297775017012E-006	2.99
			40	1.398475833157788E-007	1.97	5.076183265380021E-007	3.11
	$\alpha=0.5$	$P^0$	5	2.244975493970608E-003	-	4.553595493445027E-003	-
			10	6.694913740987843E-004	1.75	1.433071024540420E-003	1.66
			20	2.851933816998400E-004	1.23	6.606969556207020E-004	1.12
			40	1.367226188180968E-004	1.06	3.275293007699463E-004	1.01
		$P^1$	5	2.970077416293948E-004	-	7.150618409357242E-004	-
			10	1.270399125198733E-004	1.23	3.980425972542222E-004	0.85
			20	5.093888725451710E-005	1.32	1.755280846267336E-004	1.18
			40	1.627971177773421E-005	1.65	6.098150580908531E-005	1.53
		$P^2$	5	4.738891552206912E-005	-	2.323268736959109E-004	-
			10	5.248488428614583E-006	3.17	3.497357272207592E-005	2.73
			20	5.350808403013323E-007	3.29	4.548739501698698E-006	2.94
			40	6.934968715889372E-008	2.95	6.122329160500268E-007	2.89
	$\alpha=0.7$	$P^0$	5	2.226364987023399E-003	-	4.528968446397434E-003	-
			10	6.681909572138881E-004	1.73	1.429260382731581E-003	1.66
			20	2.850422396157703E-004	1.23	6.608559171080138E-004	1.11
			40	1.367054729489756E-004	1.06	3.278432462824107E-004	1.01
		$P^1$	5	2.965586452297552E-004	-	7.149566598926194E-004	-
			10	1.269543669818585E-004	1.22	3.978234263835360E-004	0.85
			20	5.092673020318355E-005	1.32	1.752325021227329E-004	1.18
			40	1.628197579228578E-005	1.65	6.068249513503552E-005	1.53
		$P^2$	5	4.736816730385127E-005	-	2.325522068324162E-004	-
			10	5.258924978488712E-006	3.17	3.526923646273578E-005	2.72
			20	6.389205641380220E-007	3.04	4.847842978104736E-006	2.86
			40	8.501389966985875E-008	2.91	6.585375965448209E-007	2.88

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Table 8. Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.2$, $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.2$	$\alpha=0.3$	$P^0$	5	1.517845954000122E-003	-	3.511606734885603E-003	-
			10	6.227976716020462E-004	1.28	1.670434180879555E-003	1.07
			20	2.917321657674542E-004	1.09	8.185201216153321E-004	1.02
			40	1.431224670285340E-004	1.02	4.068120471581059E-004	1.01
		$P^1$	5	3.242733095936642E-004	-	1.031247407943849E-003	-
			10	9.642143896521544E-005	1.75	4.664783993685712E-004	1.14
			20	2.638283335064153E-005	1.87	1.497589900759262E-004	1.64
			40	6.784781440645293E-006	1.96	4.208948155767745E-005	1.83
		$P^2$	5	4.371291470422284E-005	-	2.205191157852509E-004	-
			10	4.737332465461207E-006	3.21	5.279340967235548E-005	2.06
			20	5.836666419066179E-007	3.02	6.846287374035127E-006	2.95
			40	1.451353937788389E-007	2.01	7.721987122251807E-007	3.14
	$\alpha=0.5$	$P^0$	5	1.511832756597273E-003	-	3.502121979914235E-003	-
			10	6.220893257475209E-004	1.28	1.670078148163840E-003	1.06
			20	2.916417957441209E-004	1.09	8.186795059360371E-004	1.03
			40	1.431115461417255E-004	1.03	4.069902921658920E-004	1.01
		$P^1$	5	3.241445818441831E-004	-	1.031010873877333E-003	-
			10	9.640915416227526E-005	1.75	4.663237471849166E-004	1.14
			20	2.638161337326710E-005	1.87	1.496019341579287E-004	1.64
			40	6.783621296347129E-006	1.96	4.193227445160367E-005	1.83
		$P^2$	5	4.370398829079597E-005	-	2.206503138086216E-004	-
			10	4.735605756347349E-006	3.21	5.295075267031514E-005	2.06
			20	5.708884776768829E-007	3.05	7.003847236448262E-006	2.92
			40	7.947531188294052E-008	2.84	9.294866692183973E-007	2.91
	$\alpha=0.7$	$P^0$	5	1.505710253623656E-003	-	3.492298886676585E-003	-
			10	6.213745175059577E-004	1.28	1.669866295206499E-003	1.06
			20	2.915517673954438E-004	1.09	8.189868083813912E-004	1.03
			40	1.431015120142753E-004	1.03	4.073153866903747E-004	1.01
		$P^1$	5	3.240154726586564E-004	-	1.030637501521249E-003	-
			10	9.639741117415519E-005	1.75	4.660275194837704E-004	1.15
			20	2.638293775032076E-005	1.87	1.493034254027140E-004	1.64
			40	6.792517114587242E-006	1.96	4.163370092550497E-005	1.84
		$P^2$	5	4.369666230527429E-005	-	2.209276622101592E-004	-
			10	4.748293818147065E-006	3.20	5.325073483141655E-005	2.05
			20	6.692039201471664E-007	2.83	7.303070295557957E-006	2.87
			40	8.581080963134642E-008	2.96	9.228188670033924E-007	2.98

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Table 9. Spatial accuracy test on nonuniform meshes with generalized alternating numerical fluxes when $\rho = 1, \delta = 0.6$ $\gamma = 2, M = 10^3, T = 1$

$\delta$	$\alpha$	$P^k$	$N$	$L^2$-error	order	$L^\infty$-error	order
$\delta=0.6$	$\alpha=0.3$	$P^0$	5	2.218435560209597E-003	-	4.988165866475595E-003	-
			10	7.026515160120938E-004	1.66	1.615537336821858E-003	1.63
			20	3.238249412680320E-004	1.12	9.344015225186842E-004	0.79
			40	1.711992658914959E-004	0.92	5.144905069215206E-004	0.86
		$P^1$	5	3.213669677505128E-004	-	6.232490424656974E-004	-
			10	1.281866059118811E-004	1.33	3.003107660959630E-004	1.05
			20	5.143385599212686E-005	1.32	1.603700097095964E-004	0.91
			40	1.645991800902195E-005	1.64	5.811184200627087E-005	1.46
		$P^2$	5	5.667492210358004E-005	-	3.505324110815492E-004	-
			10	5.578235035051399E-006	3.17	4.414888437790894E-005	2.99
			20	7.175666478549729E-007	3.26	5.738400869813071E-006	2.94
			40	1.571281022897026E-007	2.19	8.425543243264881E-007	2.77
	$\alpha=0.5$	$P^0$	5	2.200335979338480E-003	-	4.956964974689847E-003	-
			10	7.011795560405282E-004	1.64	1.611200638415444E-003	1.62
			20	3.235547410371371E-004	1.12	9.339361814586587E-004	0.79
			40	1.711432886897817E-004	0.92	5.145417733177721E-004	0.86
		$P^1$	5	3.208864422126183E-004	-	6.227158645046031E-004	-
			10	1.280994719241301E-004	1.32	3.001105667075760E-004	1.05
			20	5.142050916436299E-005	1.32	1.602085136691192E-004	0.91
			40	1.645805875671424E-005	1.64	5.795371024010622E-005	1.47
		$P^2$	5	5.664565316990849E-005	-	3.502258146088582E-004	-
			10	5.576080115885499E-006	3.34	4.398437752902102E-005	2.99
			20	7.071821450175616E-007	2.98	5.873580067984358E-006	2.90
			40	9.970460744355951E-008	2.83	8.509068596339163E-007	2.79
	$\alpha=0.7$	$P^0$	5	2.181711723758149E-003	-	4.924677437389718E-003	-
			10	6.996904701082016E-004	1.63	1.606649219131593E-003	1.62
			20	3.232837185701581E-004	1.11	9.336145912944857E-004	0.78
			40	1.710880587147894E-004	0.92	5.147405087257530E-004	0.86
		$P^1$	5	3.203996862436910E-004	-	6.223220958377807E-004	-
			10	1.280113679639955E-004	1.32	2.997662559044798E-004	1.05
			20	5.140813415494180E-005	1.32	1.599050123649873E-004	0.91
			40	1.646026145542672E-005	1.64	5.765402843328131E-005	1.47
		$P^2$	5	5.661762164027293E-005	-	3.497734484754480E-004	-
			10	5.586181252970688E-006	3.34	4.367727087437284E-005	3.00
			20	7.886608314227983E-007	2.82	6.172794085018479E-006	2.82
			40	9.631333432841155E-008	3.03	1.149606829718052E-006	2.42

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