On the absolute stability of a two-step third order method on a graded mesh for an initial-value problem
A two-step third order method on a variable mesh for the approximation of nonlinear IVP: u ″ = f ( t , u , u ′ ) , u ( t 0 ) = γ 0 , u ′ ( t 0 ) = γ 1 is proposed. For computation, only a monotonically decreasing mesh will be employed. The method when applied to a test equation u ″ + 2 α u ′ + β 2 u...
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| Published in | Computational & applied mathematics Vol. 40; no. 1 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.02.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2238-3603 1807-0302 |
| DOI | 10.1007/s40314-021-01416-7 |
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| Abstract | A two-step third order method on a variable mesh for the approximation of nonlinear IVP:
u
″
=
f
(
t
,
u
,
u
′
)
,
u
(
t
0
)
=
γ
0
,
u
′
(
t
0
)
=
γ
1
is proposed. For computation, only a monotonically decreasing mesh will be employed. The method when applied to a test equation
u
″
+
2
α
u
′
+
β
2
u
=
g
(
t
)
,
α
>
β
≥
0
,
is shown to be unconditionally stable. The proposed method is applicable to solve singular problems. A special technique is required to compute the method near the singular point. Several problems of physical significance including three problems on boundary layer are examined to illustrate the convergent character and usefulness of the approximation. Approximate solutions are provided to validate the functionality of the suggested approximation. |
|---|---|
| AbstractList | A two-step third order method on a variable mesh for the approximation of nonlinear IVP:
u
″
=
f
(
t
,
u
,
u
′
)
,
u
(
t
0
)
=
γ
0
,
u
′
(
t
0
)
=
γ
1
is proposed. For computation, only a monotonically decreasing mesh will be employed. The method when applied to a test equation
u
″
+
2
α
u
′
+
β
2
u
=
g
(
t
)
,
α
>
β
≥
0
,
is shown to be unconditionally stable. The proposed method is applicable to solve singular problems. A special technique is required to compute the method near the singular point. Several problems of physical significance including three problems on boundary layer are examined to illustrate the convergent character and usefulness of the approximation. Approximate solutions are provided to validate the functionality of the suggested approximation. A two-step third order method on a variable mesh for the approximation of nonlinear IVP: u″=f(t,u,u′),u(t0)=γ0,u′(t0)=γ1 is proposed. For computation, only a monotonically decreasing mesh will be employed. The method when applied to a test equation u″+2αu′+β2u=g(t),α>β≥0, is shown to be unconditionally stable. The proposed method is applicable to solve singular problems. A special technique is required to compute the method near the singular point. Several problems of physical significance including three problems on boundary layer are examined to illustrate the convergent character and usefulness of the approximation. Approximate solutions are provided to validate the functionality of the suggested approximation. |
| ArticleNumber | 35 |
| Author | McKee, Sean Ghosh, Bishnu Pada Mohanty, R. K. |
| Author_xml | – sequence: 1 givenname: R. K. orcidid: 0000-0001-6832-1239 surname: Mohanty fullname: Mohanty, R. K. email: rmohanty@sau.ac.in organization: Department of Applied Mathematics, South Asian University – sequence: 2 givenname: Bishnu Pada surname: Ghosh fullname: Ghosh, Bishnu Pada organization: Department of Applied Mathematics, South Asian University – sequence: 3 givenname: Sean surname: McKee fullname: McKee, Sean organization: Department of Mathematics and Statistics, University of Strathclyde |
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| Cites_doi | 10.1137/1.9781611970944 10.1016/S0893-9659(04)90019-5 10.1016/S0377-0427(96)00132-X 10.1007/BF01401041 10.1201/9780203908518 10.1093/imanum/23.2.197 10.1016/j.cam.2004.11.045 10.1016/0377-0427(94)00127-8 10.1007/BF01933627 10.1023/A:1026167824656 10.1080/00207160801965271 10.1016/j.aml.2015.01.008 10.1080/00207160701871843 10.1016/0377-0427(85)90018-4 10.1016/0045-7825(84)90009-4 10.1007/s00366-019-00857-3 10.1016/j.amc.2006.05.071 10.1007/BF01931689 10.1093/imamat/18.2.189 10.1016/j.amc.2004.07.002 |
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| Keywords | 65L05 65L06 Graded mesh Nonlinear IVPs Region of absolute stability 65L07 Damped wave equation Mathematics Subject Classification Boundary layer problems Singular problem |
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| References | Rai, Ananthakrishnaiah (CR20) 1996; 67 Hairer (CR6) 1979; 32 Lambert (CR11) 1991 Chawla (CR1) 1983; 23 Chawla (CR2) 1985; 12 Jain, Iyengar, Subramanyam (CR7) 1984; 42 Mohanty (CR15) 2005; 182 Samarskii (CR23) 2001 CR8 Simos, Famelis, Tsitouras (CR24) 2003; 34 Lambert (CR10) 1973 Dahlquist (CR4) 1978; 18 Rai, Ananthakrishnaiah (CR21) 1997; 79 Mohanty (CR13) 2004; 17 Mohanty (CR14) 2005; 165 Mohanty (CR17) 2009; 86 Mohanty, Ghosh (CR18) 2021 Kelly (CR9) 1995 Hageman, Young (CR5) 2004 Mohanty (CR16) 2006; 183 Saldanha, Saldanha (CR22) 2009; 86 Coleman (CR3) 2003; 23 Mohanty, McKee (CR19) 2015; 45 Lambert, Watson (CR12) 1976; 18 AS Rai (1416_CR20) 1996; 67 RK Mohanty (1416_CR18) 2021 G Saldanha (1416_CR22) 2009; 86 JP Coleman (1416_CR3) 2003; 23 RK Mohanty (1416_CR16) 2006; 183 RK Mohanty (1416_CR19) 2015; 45 AA Samarskii (1416_CR23) 2001 RK Mohanty (1416_CR17) 2009; 86 LA Hageman (1416_CR5) 2004 MK Jain (1416_CR7) 1984; 42 AS Rai (1416_CR21) 1997; 79 MM Chawla (1416_CR2) 1985; 12 JD Lambert (1416_CR12) 1976; 18 TE Simos (1416_CR24) 2003; 34 1416_CR8 RK Mohanty (1416_CR13) 2004; 17 MM Chawla (1416_CR1) 1983; 23 RK Mohanty (1416_CR15) 2005; 182 G Dahlquist (1416_CR4) 1978; 18 E Hairer (1416_CR6) 1979; 32 JD Lambert (1416_CR11) 1991 RK Mohanty (1416_CR14) 2005; 165 JD Lambert (1416_CR10) 1973 CT Kelly (1416_CR9) 1995 |
| References_xml | – year: 2004 ident: CR5 publication-title: Applied iterative methods – year: 1995 ident: CR9 publication-title: Iterative methods for linear and non-linear equations doi: 10.1137/1.9781611970944 – volume: 17 start-page: 101 year: 2004 end-page: 105 ident: CR13 article-title: An unconditionally stable difference scheme for the one space dimensional linear hyperbolic equation publication-title: Appl Math Lett doi: 10.1016/S0893-9659(04)90019-5 – volume: 79 start-page: 167 year: 1997 end-page: 182 ident: CR21 article-title: Obrechkoff methods having additional parameters for general second order differential equations publication-title: J Comput Appl Math doi: 10.1016/S0377-0427(96)00132-X – volume: 32 start-page: 373 year: 1979 end-page: 379 ident: CR6 article-title: Unconditionally stable methods for second order differential equations publication-title: Numer Math doi: 10.1007/BF01401041 – year: 2001 ident: CR23 publication-title: The theory of difference schemes doi: 10.1201/9780203908518 – volume: 23 start-page: 197 year: 2003 end-page: 220 ident: CR3 article-title: Order conditions for a class of two-step methods for y"=f(x, y) publication-title: IMA J Numer Anal doi: 10.1093/imanum/23.2.197 – volume: 182 start-page: 173 year: 2005 end-page: 187 ident: CR15 article-title: A family of variable mesh methods for the estimates of ( ) and the solution of nonlinear two point boundary value problems with singularity publication-title: J Comput Appl Math doi: 10.1016/j.cam.2004.11.045 – volume: 67 start-page: 271 year: 1996 end-page: 276 ident: CR20 article-title: Additive parameters methods for the numerical integration of y"=f(x, y, y') publication-title: J Comput Appl Math doi: 10.1016/0377-0427(94)00127-8 – volume: 23 start-page: 541 year: 1983 end-page: 542 ident: CR1 article-title: Unconditionally stable Numerov type methods for second order differential equations publication-title: BIT doi: 10.1007/BF01933627 – year: 1973 ident: CR10 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discretization for the initial-value problem on a graded mesh publication-title: Eng Computs doi: 10.1007/s00366-019-00857-3 – ident: CR8 – volume: 183 start-page: 477 year: 2006 end-page: 485 ident: CR16 article-title: A class of non-uniform mesh three point arithmetic average discretization for y"=f(x, y, y') and the estimates of y' publication-title: Appl Math Comput doi: 10.1016/j.amc.2006.05.071 – volume: 18 start-page: 133 year: 1978 end-page: 136 ident: CR4 article-title: On accuracy and unconditional stability of linear multistep methods for second order differential equations publication-title: BIT doi: 10.1007/BF01931689 – volume: 18 start-page: 189 year: 1976 end-page: 202 ident: CR12 article-title: Symmetric multistep methods for periodic initial value problems J publication-title: Inst Maths Applics doi: 10.1093/imamat/18.2.189 – year: 1991 ident: CR11 publication-title: Numerical methods for ordinary differential systems: the initial value problem – volume: 18 start-page: 133 year: 1978 ident: 1416_CR4 publication-title: BIT doi: 10.1007/BF01931689 – volume: 18 start-page: 189 year: 1976 ident: 1416_CR12 publication-title: Inst Maths Applics doi: 10.1093/imamat/18.2.189 – volume: 17 start-page: 101 year: 2004 ident: 1416_CR13 publication-title: Appl Math Lett doi: 10.1016/S0893-9659(04)90019-5 – volume-title: The theory of difference schemes year: 2001 ident: 1416_CR23 doi: 10.1201/9780203908518 – volume-title: Applied iterative methods year: 2004 ident: 1416_CR5 – volume: 42 start-page: 273 year: 1984 ident: 1416_CR7 publication-title: Comput Methods Appl Mech Eng doi: 10.1016/0045-7825(84)90009-4 – volume: 23 start-page: 197 year: 2003 ident: 1416_CR3 publication-title: IMA J Numer Anal doi: 10.1093/imanum/23.2.197 – year: 2021 ident: 1416_CR18 publication-title: Eng Computs doi: 10.1007/s00366-019-00857-3 – volume: 12 start-page: 217 year: 1985 ident: 1416_CR2 publication-title: Comput Appl Math doi: 10.1016/0377-0427(85)90018-4 – volume-title: Iterative methods for linear and non-linear equations year: 1995 ident: 1416_CR9 doi: 10.1137/1.9781611970944 – volume: 165 start-page: 229 year: 2005 ident: 1416_CR14 publication-title: Appl Math Comput doi: 10.1016/j.amc.2004.07.002 – ident: 1416_CR8 – volume: 34 start-page: 27 year: 2003 ident: 1416_CR24 publication-title: Numer Algorithms doi: 10.1023/A:1026167824656 – volume: 183 start-page: 477 year: 2006 ident: 1416_CR16 publication-title: Appl Math Comput doi: 10.1016/j.amc.2006.05.071 – volume: 79 start-page: 167 year: 1997 ident: 1416_CR21 publication-title: J Comput Appl Math doi: 10.1016/S0377-0427(96)00132-X – volume: 23 start-page: 541 year: 1983 ident: 1416_CR1 publication-title: BIT doi: 10.1007/BF01933627 – volume-title: Computational methods in ordinary differential equations year: 1973 ident: 1416_CR10 – volume: 45 start-page: 31 year: 2015 ident: 1416_CR19 publication-title: Appl Math Lett doi: 10.1016/j.aml.2015.01.008 – volume: 67 start-page: 271 year: 1996 ident: 1416_CR20 publication-title: J Comput Appl Math doi: 10.1016/0377-0427(94)00127-8 – volume: 182 start-page: 173 year: 2005 ident: 1416_CR15 publication-title: J Comput Appl Math doi: 10.1016/j.cam.2004.11.045 – volume: 32 start-page: 373 year: 1979 ident: 1416_CR6 publication-title: Numer Math doi: 10.1007/BF01401041 – volume-title: Numerical methods for ordinary differential systems: the initial value problem year: 1991 ident: 1416_CR11 – volume: 86 start-page: 2061 year: 2009 ident: 1416_CR17 publication-title: Int J Comput Math doi: 10.1080/00207160801965271 – volume: 86 start-page: 1424 year: 2009 ident: 1416_CR22 publication-title: Int J Comput Math doi: 10.1080/00207160701871843 |
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| Snippet | A two-step third order method on a variable mesh for the approximation of nonlinear IVP:
u
″
=
f
(
t
,
u
,
u
′
)
,
u
(
t
0
)
=
γ
0
,
u
′
(
t
0
)
=
γ
1
is... A two-step third order method on a variable mesh for the approximation of nonlinear IVP: u″=f(t,u,u′),u(t0)=γ0,u′(t0)=γ1 is proposed. For computation, only a... |
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| SubjectTerms | Applications of Mathematics Applied physics Approximation Boundary layers Computational mathematics Computational Mathematics and Numerical Analysis Mathematical Applications in Computer Science Mathematical Applications in the Physical Sciences Mathematics Mathematics and Statistics |
| Title | On the absolute stability of a two-step third order method on a graded mesh for an initial-value problem |
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