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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_pde_1d_parab_keller (d03pe)

Purpose

nag_pde_1d_parab_keller (d03pe) integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a Backward Differentiation Formula (BDF) method.

Syntax

[ts, u, rsave, isave, ind, ifail] = d03pe(ts, tout, pdedef, bndary, u, x, nleft, acc, rsave, isave, itask, itrace, ind, 'npde', npde, 'npts', npts)
[ts, u, rsave, isave, ind, ifail] = nag_pde_1d_parab_keller(ts, tout, pdedef, bndary, u, x, nleft, acc, rsave, isave, itask, itrace, ind, 'npde', npde, 'npts', npts)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22: lrsave, lisave have been removed from the interface
.

Description

nag_pde_1d_parab_keller (d03pe) integrates the system of first-order PDEs
Gi(x,t,U,Ux,Ut) = 0,  i = 1,2,,npde.
Gi(x,t,U,Ux,Ut)=0,  i=1,2,,npde.
(1)
In particular the functions GiGi must have the general form
npde
Gi = Pi,j(Uj)/(t) + Qi,  i = 1,2,,npde,  axb,tt0,
j = 1
Gi=j=1npdePi,j Uj t +Qi,  i=1,2,,npde,  axb,tt0,
(2)
where Pi,jPi,j and QiQi depend on xx, tt, UU, UxUx and the vector UU is the set of solution values
U (x,t) = [ U1 (x,t) ,, Unpde (x,t) ]T ,
U (x,t) = [ U 1 (x,t) ,, U npde (x,t) ] T ,
(3)
and the vector UxUx is its partial derivative with respect to xx. Note that Pi,jPi,j and QiQi must not depend on (U)/(t) U t .
The integration in time is from t0t0 to touttout, over the space interval axbaxb, where a = x1a=x1 and b = xnptsb=xnpts are the leftmost and rightmost points of a user-defined mesh x1,x2,,xnptsx1,x2,,xnpts. The mesh should be chosen in accordance with the expected behaviour of the solution.
The PDE system which is defined by the functions GiGi must be specified in pdedef.
The initial values of the functions U(x,t)U(x,t) must be given at t = t0t=t0. For a first-order system of PDEs, only one boundary condition is required for each PDE component UiUi. The npde boundary conditions are separated into nana at the left-hand boundary x = ax=a, and nbnb at the right-hand boundary x = bx=b, such that na + nb = npdena+nb=npde. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of UiUi at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for UiUi should be specified at the left-hand boundary. Incorrect positioning of boundary conditions generally results in initialization or integration difficulties in the underlying time integration functions.
The boundary conditions have the form:
GiL(x,t,U,Ut) = 0  at ​x = a,  i = 1,2,,na
GiL(x,t,U,Ut)=0  at ​x=a,  i=1,2,,na
(4)
at the left-hand boundary, and
GiR(x,t,U,Ut) = 0  at ​x = b,  i = 1,2,,nb
GiR(x,t,U,Ut)=0  at ​x=b,  i=1,2,,nb
(5)
at the right-hand boundary.
Note that the functions GiLGiL and GiRGiR must not depend on UxUx, since spatial derivatives are not determined explicitly in the Keller box scheme (see Keller (1970)). If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that GiLGiL and GiRGiR must be linear with respect to time derivatives, so that the boundary conditions have the general form
npde
Ei,jL(Uj)/(t) + SiL = 0,  i = 1,2,,na
j = 1
j=1npdeEi,jL Uj t +SiL=0,   i=1,2,,na
(6)
at the left-hand boundary, and
npde
Ei,jR(Uj)/(t) + SiR = 0,  i = 1,2,,nb
j = 1
j=1npdeEi,jR Uj t +SiR=0,   i=1,2,,nb
(7)
at the right-hand boundary, where Ei,jLEi,jL, Ei,jREi,jR, SiLSiL, and SiRSiR depend on xx, tt and UU only.
The boundary conditions must be specified in bndary.
The problem is subject to the following restrictions:
(i) t0 < toutt0<tout, so that integration is in the forward direction;
(ii) Pi,jPi,j and QiQi must not depend on any time derivatives;
(iii) The evaluation of the function GiGi is done at the mid-points of the mesh intervals by calling the pdedef for each mid-point in turn. Any discontinuities in the function must therefore be at one or more of the mesh points x1,x2,,xnptsx1,x2,,xnpts;
(iv) At least one of the functions Pi,jPi,j must be nonzero so that there is a time derivative present in the problem.
In this method of lines approach the Keller box scheme (see Keller (1970)) is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of UiUi at each mesh point. In total there are npde × nptsnpde×npts ODEs in the time direction. This system is then integrated forwards in time using a BDF method.

References

Berzins M (1990) Developments in the NAG Library software for parabolic equations Scientific Software Systems (eds J C Mason and M G Cox) 59–72 Chapman and Hall
Berzins M, Dew P M and Furzeland R M (1989) Developing software for time-dependent problems using the method of lines and differential-algebraic integrators Appl. Numer. Math. 5 375–397
Keller H B (1970) A new difference scheme for parabolic problems Numerical Solutions of Partial Differential Equations (ed J Bramble) 2 327–350 Academic Press
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99

Parameters

Compulsory Input Parameters

1:     ts – double scalar
The initial value of the independent variable tt.
Constraint: ts < toutts<tout.
2:     tout – double scalar
The final value of tt to which the integration is to be carried out.
3:     pdedef – function handle or string containing name of m-file
pdedef must compute the functions GiGi which define the system of PDEs. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_keller (d03pe).
[res, ires] = pdedef(npde, t, x, u, ut, ux, ires)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable tt.
3:     x – double scalar
The current value of the space variable xx.
4:     u(npde) – double array
u(i)ui contains the value of the component Ui(x,t)Ui(x,t), for i = 1,2,,npdei=1,2,,npde.
5:     ut(npde) – double array
ut(i)uti contains the value of the component (Ui(x,t))/(t) Ui(x,t) t , for i = 1,2,,npdei=1,2,,npde.
6:     ux(npde) – double array
ux(i)uxi contains the value of the component (Ui(x,t))/(x) Ui(x,t) x , for i = 1,2,,npdei=1,2,,npde.
7:     ires – int64int32nag_int scalar
The form of GiGi that must be returned in the array res.
ires = 1ires=-1
Equation (8) must be used.
ires = 1ires=1
Equation (9) must be used.
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=3
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ires = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_keller (d03pe) returns to the calling function with the error indicator set to ifail = 4ifail=4.

Output Parameters

1:     res(npde) – double array
res(i)resi must contain the iith component of GG, for i = 1,2,,npdei=1,2,,npde, where GG is defined as
npde
Gi = Pi,j(Uj)/(t),
j = 1
Gi=j=1npdePi,j Uj t ,
(8)
i.e., only terms depending explicitly on time derivatives, or
npde
Gi = Pi,j(Uj)/(t) + Qi,
j = 1
Gi=j=1npdePi,j Uj t +Qi,
(9)
i.e., all terms in equation (2).
The definition of GG is determined by the input value of ires.
2:     ires – int64int32nag_int scalar
Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions, as described below:
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=3
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ires = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_keller (d03pe) returns to the calling function with the error indicator set to ifail = 4ifail=4.
4:     bndary – function handle or string containing name of m-file
bndary must compute the functions GiLGiL and GiRGiR which define the boundary conditions as in equations (4) and (5).
[res, ires] = bndary(npde, t, ibnd, nobc, u, ut, ires)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable tt.
3:     ibnd – int64int32nag_int scalar
Determines the position of the boundary conditions.
ibnd = 0ibnd=0
bndary must compute the left-hand boundary condition at x = ax=a.
ibnd0ibnd0
Indicates that bndary must compute the right-hand boundary condition at x = bx=b.
4:     nobc – int64int32nag_int scalar
Specifies the number of boundary conditions at the boundary specified by ibnd.
5:     u(npde) – double array
u(i)ui contains the value of the component Ui(x,t)Ui(x,t) at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
6:     ut(npde) – double array
ut(i)uti contains the value of the component (Ui(x,t))/(t) Ui(x,t) t  at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
7:     ires – int64int32nag_int scalar
The form GiLGiL (or GiRGiR) that must be returned in the array res.
ires = 1ires=-1
Equation (10) must be used.
ires = 1ires=1
Equation (11) must be used.

Output Parameters

1:     res(nobc) – double array
res(i)resi must contain the iith component of GLGL or GRGR, depending on the value of ibnd, for i = 1,2,,nobci=1,2,,nobc, where GLGL is defined as
npde
GiL = Ei,jL(Uj)/(t),
j = 1
GiL=j=1npde Ei,jL Uj t ,
(10)
i.e., only terms depending explicitly on time derivatives, or
npde
GiL = Ei,jL(Uj)/(t) + SiL,
j = 1
GiL=j=1npde Ei,jL Uj t +SiL,
(11)
i.e., all terms in equation (6), and similarly for GiRGiR.
The definitions of GLGL and GRGR are determined by the input value of ires.
2:     ires – int64int32nag_int scalar
Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions, as described below:
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=3
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ires = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_keller (d03pe) returns to the calling function with the error indicator set to ifail = 4ifail=4.
5:     u(npde,npts) – double array
npde, the first dimension of the array, must satisfy the constraint npde1npde1.
The initial values of U(x,t)U(x,t) at t = tst=ts and the mesh points x(j)xj, for j = 1,2,,nptsj=1,2,,npts.
6:     x(npts) – double array
npts, the dimension of the array, must satisfy the constraint npts3npts3.
The mesh points in the spatial direction. x(1)x1 must specify the left-hand boundary, aa, and x(npts)xnpts must specify the right-hand boundary, bb.
Constraint: x(1) < x(2) < < x(npts)x1<x2<<xnpts.
7:     nleft – int64int32nag_int scalar
The number nana of boundary conditions at the left-hand mesh point x(1)x1.
Constraint: 0nleftnpde0nleftnpde.
8:     acc – double scalar
A positive quantity for controlling the local error estimate in the time integration. If E(i,j)E(i,j) is the estimated error for UiUi at the jjth mesh point, the error test is:
|E(i,j)| = acc × (1.0 + |u(i,j)|).
|E(i,j)|=acc×(1.0+|uij|).
Constraint: acc > 0.0acc>0.0.
9:     rsave(lrsave) – double array
lrsave, the dimension of the array, must satisfy the constraint lrsave(4 × npde + nleft + 14) × npde × npts + (3 × npde + 21) × npde +  7 × npts + 54lrsave(4×npde+nleft+14)×npde×npts+(3×npde+21)×npde+7×npts+54.
If ind = 0ind=0, rsave need not be set on entry.
If ind = 1ind=1, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
10:   isave(lisave) – int64int32nag_int array
lisave, the dimension of the array, must satisfy the constraint lisavenpde × npts + 24lisavenpde×npts+24.
If ind = 0ind=0, isave need not be set on entry.
If ind = 1ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
isave(1)isave1
Contains the number of steps taken in time.
isave(2)isave2
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
isave(3)isave3
Contains the number of Jacobian evaluations performed by the time integrator.
isave(4)isave4
Contains the order of the last backward differentiation formula method used.
isave(5)isave5
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LULU decomposition of the Jacobian matrix.
11:   itask – int64int32nag_int scalar
Specifies the task to be performed by the ODE integrator.
itask = 1itask=1
Normal computation of output values uu at t = toutt=tout.
itask = 2itask=2
Take one step and return.
itask = 3itask=3
Stop at the first internal integration point at or beyond t = toutt=tout.
Constraint: itask = 1itask=1, 22 or 33.
12:   itrace – int64int32nag_int scalar
The level of trace information required from nag_pde_1d_parab_keller (d03pe) and the underlying ODE solver as follows:
itrace1itrace-1
No output is generated.
itrace = 0itrace=0
Only warning messages from the PDE solver are printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
itrace = 1itrace=1
Output from the underlying ODE solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
itrace = 2itrace=2
Output from the underlying ODE solver is similar to that produced when itrace = 1itrace=1, except that the advisory messages are given in greater detail.
itrace3itrace3
Output from the underlying ODE solver is similar to that produced when itrace = 2itrace=2, except that the advisory messages are given in greater detail.
You are advised to set itrace = 0itrace=0, unless you are experienced with sub-chapter D02M–N.
13:   ind – int64int32nag_int scalar
Indicates whether this is a continuation call or a new integration.
ind = 0ind=0
Starts or restarts the integration in time.
ind = 1ind=1
Continues the integration after an earlier exit from the function. In this case, only the parameters tout and ifail should be reset between calls to nag_pde_1d_parab_keller (d03pe).
Constraint: ind = 0ind=0 or 11.

Optional Input Parameters

1:     npde – int64int32nag_int scalar
Default: The first dimension of the array u.
The number of PDEs in the system to be solved.
Constraint: npde1npde1.
2:     npts – int64int32nag_int scalar
Default: The dimension of the array x and the second dimension of the array u. (An error is raised if these dimensions are not equal.)
The number of mesh points in the interval [a,b][a,b].
Constraint: npts3npts3.

Input Parameters Omitted from the MATLAB Interface

lrsave lisave

Output Parameters

1:     ts – double scalar
The value of tt corresponding to the solution values in u. Normally ts = toutts=tout.
2:     u(npde,npts) – double array
u(i,j)uij will contain the computed solution at t = tst=ts.
3:     rsave(lrsave) – double array
If ind = 1ind=1, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
4:     isave(lisave) – int64int32nag_int array
If ind = 1ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
isave(1)isave1
Contains the number of steps taken in time.
isave(2)isave2
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
isave(3)isave3
Contains the number of Jacobian evaluations performed by the time integrator.
isave(4)isave4
Contains the order of the last backward differentiation formula method used.
isave(5)isave5
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LULU decomposition of the Jacobian matrix.
5:     ind – int64int32nag_int scalar
ind = 1ind=1.
6:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
On entry,touttstoutts,
or(toutts)(tout-ts) is too small,
oritask1itask1, 22 or 33,
ormesh points ​x(i)mesh points ​xi are not ordered correctly,
ornpts < 3npts<3,
ornpde < 1npde<1,
ornleft is not in the range 00 to npde,
oracc0.0acc0.0,
orind0ind0 or 11,
orlrsave is too small,
orlisave is too small,
ornag_pde_1d_parab_keller (d03pe) called initially with ind = 1ind=1.
W ifail = 2ifail=2
The underlying ODE solver cannot make any further progress across the integration range from the current point t = tst=ts with the supplied value of acc. The components of u contain the computed values at the current point t = tst=ts.
W ifail = 3ifail=3
In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or acc is too small for the integration to continue. Incorrect positioning of boundary conditions may also result in this error. Integration was successful as far as t = tst=ts.
  ifail = 4ifail=4
In setting up the ODE system, the internal initialization function was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to 33 in the pdedef or bndary, when the residual in the underlying ODE solver was being evaluated. Incorrect positioning of boundary conditions may also result in this error.
  ifail = 5ifail=5
In solving the ODE system, a singular Jacobian has been encountered. You should check their problem formulation.
W ifail = 6ifail=6
When evaluating the residual in solving the ODE system, ires was set to 22 in one of pdedef or bndary. Integration was successful as far as t = tst=ts.
  ifail = 7ifail=7
The value of acc is so small that the function is unable to start the integration in time.
  ifail = 8ifail=8
In either, pdedef or bndary, ires was set to an invalid value.
  ifail = 9ifail=9 (nag_ode_ivp_stiff_imp_revcom (d02nn))
A serious error has occurred in an internal call to the specified function. Check the problem specification and all parameters and array dimensions. Setting itrace = 1itrace=1 may provide more information. If the problem persists, contact NAG.
W ifail = 10ifail=10
The required task has been completed, but it is estimated that a small change in acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when itask2itask2.)
  ifail = 11ifail=11
An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current advisory message unit).

Accuracy

nag_pde_1d_parab_keller (d03pe) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy parameter, acc.

Further Comments

The Keller box scheme can be used to solve higher-order problems which have been reduced to first-order by the introduction of new variables (see the example problem in nag_pde_1d_parab_dae_keller (d03pk)). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (nag_pde_1d_parab_fd (d03pc) or nag_pde_1d_parab_dae_fd (d03ph) for example), but at the expense of increased CPU time due to the larger number of function evaluations required.
It should be noted that the Keller box scheme, in common with other central-difference schemes, may be unsuitable for some hyperbolic first-order problems such as the apparently simple linear advection equation Ut + aUx = 0Ut+aUx=0, where aa is a constant, resulting in spurious oscillations due to the lack of dissipation. This type of problem requires a discretization scheme with upwind weighting (nag_pde_1d_parab_convdiff (d03pf) for example), or the addition of a second-order artificial dissipation term.
The time taken depends on the complexity of the system and on the accuracy requested.

Example

function nag_pde_1d_parab_keller_example
ts = 0;
tout = 0.2;
u = [1, 1.025315120524429, 1.051271096376024, 1.077884150884632, ...
    1.105170918075648, 1.133148453066826, 1.161834242728283, ...
    1.191246216612358, 1.22140275816017, 1.252322716191864, ...
    1.284025416687741, 1.316530674867622, 1.349858807576003, ...
    1.384030645980751, 1.419067548593257, 1.454991414618201, ...
    1.49182469764127, 1.529590419663379, 1.568312185490169, ...
    1.608014197485783, 1.648721270700128, 1.690458848379091, ...
    1.733253017867395, 1.777130526914038, 1.822118800390509, ...
    1.868245957432222, 1.915540829013896, 1.964032975969847, ...
    2.013752707470477, 2.064731099966486, 2.117000016612675, ...
    2.170592127183443, 2.225540928492468, 2.281880765329304, ...
    2.339646851925991, 2.398875293967098, 2.45960311115695, ...
    2.521868260358148, 2.585709659315846, 2.651167210982607, ...
    2.718281828459045;
     0, 0.02499739591471233, 0.04997916927067833, 0.07492970727274234, ...
    0.09983341664682815, 0.1246747333852277, 0.1494381324735992, ...
    0.1741081375935959, 0.1986693307950612, 0.2231063621317455, ...
    0.2474039592545229, 0.2715469369561129, 0.2955202066613395, ...
    0.319308785857001, 0.3428978074554513, 0.3662725290860476, ...
    0.3894183423086505, 0.4123207817434247, 0.4349655341112302, ...
    0.4573384471789554, 0.479425538604203, 0.5012130046737979, ...
    0.5226872289306592, 0.5438347906836426, 0.5646424733950354, ...
    0.5850972729404622, 0.6051864057360395, 0.6248973167276999, ...
    0.644217687237691, 0.6631354426633497, 0.6816387600233341, ...
    0.6997160753466035, 0.7173560908995228, 0.7345477822465786, ...
    0.7512804051402927, 0.7675435022360271, 0.7833269096274834, ...
    0.7986207631988143, 0.8134155047893737, 0.8277018881672576, ...
    0.8414709848078965];
x = [0;
     0.025;
     0.05;
     0.075;
     0.1;
     0.125;
     0.15;
     0.175;
     0.2;
     0.225;
     0.25;
     0.275;
     0.3;
     0.325;
     0.35;
     0.375;
     0.4;
     0.425;
     0.45;
     0.475;
     0.5;
     0.525;
     0.55;
     0.575;
     0.6;
     0.625;
     0.65;
     0.675;
     0.7;
     0.725;
     0.75;
     0.775;
     0.8;
     0.825;
     0.85;
     0.875;
     0.9;
     0.925;
     0.95;
     0.975;
     1];
nleft = int64(1);
acc = 1e-06;
rsave = zeros(2281, 1);
isave = zeros(106, 1, 'int64');
itask = int64(1);
itrace = int64(0);
ind = int64(0);
[tsOut, uOut, rsaveOut, isaveOut, indOut, ifail] = ...
    nag_pde_1d_parab_keller(ts, tout, @pdedef, @bndary, u, x, nleft, acc, ...
    rsave, isave, itask, itrace, ind);
 tsOut, uOut, ifail

function [res, ires] = pdedef(npde, t, x, u, ut, ux, ires)

      if (ires == -1)
         res(1) = ut(1);
         res(2) = ut(2);
      else
         res(1) = ut(1) + ux(1) + ux(2);
         res(2) = ut(2) + 4.0d0*ux(1) + ux(2);
      end
function [res, ires] = bndary(npde, t, ibnd, nobc, u, ut, ires)
      if (ibnd == 0)
         if (ires == -1)
            res(1) = 0;
         else
            res(1) = u(1) - 0.5*(exp(t)+exp(-3*t)) - 0.25*(sin(-3*t)-sin(t));
         end
      else
         if (ires == -1)
            res(1) = 0;
         else
            res(1) = u(2) - exp(1-3*t) + exp(1+t) - 0.5*(sin(1-3*t)+sin(1+t));
         end
      end
 

tsOut =

    0.2000


uOut =

  Columns 1 through 9

    0.6943    0.7158    0.7380    0.7609    0.7845    0.8088    0.8339    0.8597    0.8864
   -0.8555   -0.8499   -0.8447   -0.8398   -0.8352   -0.8311   -0.8274   -0.8242   -0.8215

  Columns 10 through 18

    0.9138    0.9420    0.9711    1.0010    1.0318    1.0635    1.0961    1.1296    1.1640
   -0.8192   -0.8176   -0.8164   -0.8159   -0.8160   -0.8168   -0.8182   -0.8204   -0.8233

  Columns 19 through 27

    1.1995    1.2359    1.2733    1.3117    1.3512    1.3918    1.4334    1.4762    1.5201
   -0.8269   -0.8314   -0.8367   -0.8428   -0.8499   -0.8579   -0.8668   -0.8767   -0.8877

  Columns 28 through 36

    1.5652    1.6115    1.6590    1.7078    1.7578    1.8091    1.8618    1.9158    1.9713
   -0.8997   -0.9128   -0.9270   -0.9424   -0.9590   -0.9768   -0.9958   -1.0162   -1.0379

  Columns 37 through 41

    2.0281    2.0864    2.1462    2.2075    2.2703
   -1.0609   -1.0854   -1.1113   -1.1387   -1.1676


ifail =

                    0


function d03pe_example
ts = 0;
tout = 0.2;
u = [1, 1.025315120524429, 1.051271096376024, 1.077884150884632, ...
    1.105170918075648, 1.133148453066826, 1.161834242728283, ...
    1.191246216612358, 1.22140275816017, 1.252322716191864, ...
    1.284025416687741, 1.316530674867622, 1.349858807576003, ...
    1.384030645980751, 1.419067548593257, 1.454991414618201, ...
    1.49182469764127, 1.529590419663379, 1.568312185490169, ...
    1.608014197485783, 1.648721270700128, 1.690458848379091, ...
    1.733253017867395, 1.777130526914038, 1.822118800390509, ...
    1.868245957432222, 1.915540829013896, 1.964032975969847, ...
    2.013752707470477, 2.064731099966486, 2.117000016612675, ...
    2.170592127183443, 2.225540928492468, 2.281880765329304, ...
    2.339646851925991, 2.398875293967098, 2.45960311115695, ...
    2.521868260358148, 2.585709659315846, 2.651167210982607, ...
    2.718281828459045;
     0, 0.02499739591471233, 0.04997916927067833, 0.07492970727274234, ...
    0.09983341664682815, 0.1246747333852277, 0.1494381324735992, ...
    0.1741081375935959, 0.1986693307950612, 0.2231063621317455, ...
    0.2474039592545229, 0.2715469369561129, 0.2955202066613395, ...
    0.319308785857001, 0.3428978074554513, 0.3662725290860476, ...
    0.3894183423086505, 0.4123207817434247, 0.4349655341112302, ...
    0.4573384471789554, 0.479425538604203, 0.5012130046737979, ...
    0.5226872289306592, 0.5438347906836426, 0.5646424733950354, ...
    0.5850972729404622, 0.6051864057360395, 0.6248973167276999, ...
    0.644217687237691, 0.6631354426633497, 0.6816387600233341, ...
    0.6997160753466035, 0.7173560908995228, 0.7345477822465786, ...
    0.7512804051402927, 0.7675435022360271, 0.7833269096274834, ...
    0.7986207631988143, 0.8134155047893737, 0.8277018881672576, ...
    0.8414709848078965];
x = [0;
     0.025;
     0.05;
     0.075;
     0.1;
     0.125;
     0.15;
     0.175;
     0.2;
     0.225;
     0.25;
     0.275;
     0.3;
     0.325;
     0.35;
     0.375;
     0.4;
     0.425;
     0.45;
     0.475;
     0.5;
     0.525;
     0.55;
     0.575;
     0.6;
     0.625;
     0.65;
     0.675;
     0.7;
     0.725;
     0.75;
     0.775;
     0.8;
     0.825;
     0.85;
     0.875;
     0.9;
     0.925;
     0.95;
     0.975;
     1];
nleft = int64(1);
acc = 1e-06;
rsave = zeros(2281, 1);
isave = zeros(106, 1, 'int64');
itask = int64(1);
itrace = int64(0);
ind = int64(0);
[tsOut, uOut, rsaveOut, isaveOut, indOut, ifail] = ...
    d03pe(ts, tout, @pdedef, @bndary, u, x, nleft, acc, ...
    rsave, isave, itask, itrace, ind);
 tsOut, uOut, ifail

function [res, ires] = pdedef(npde, t, x, u, ut, ux, ires)

      if (ires == -1)
         res(1) = ut(1);
         res(2) = ut(2);
      else
         res(1) = ut(1) + ux(1) + ux(2);
         res(2) = ut(2) + 4.0d0*ux(1) + ux(2);
      end
function [res, ires] = bndary(npde, t, ibnd, nobc, u, ut, ires)
      if (ibnd == 0)
         if (ires == -1)
            res(1) = 0;
         else
            res(1) = u(1) - 0.5*(exp(t)+exp(-3*t)) - 0.25*(sin(-3*t)-sin(t));
         end
      else
         if (ires == -1)
            res(1) = 0;
         else
            res(1) = u(2) - exp(1-3*t) + exp(1+t) - 0.5*(sin(1-3*t)+sin(1+t));
         end
      end
 

tsOut =

    0.2000


uOut =

  Columns 1 through 9

    0.6943    0.7158    0.7380    0.7609    0.7845    0.8088    0.8339    0.8597    0.8864
   -0.8555   -0.8499   -0.8447   -0.8398   -0.8352   -0.8311   -0.8274   -0.8242   -0.8215

  Columns 10 through 18

    0.9138    0.9420    0.9711    1.0010    1.0318    1.0635    1.0961    1.1296    1.1640
   -0.8192   -0.8176   -0.8164   -0.8159   -0.8160   -0.8168   -0.8182   -0.8204   -0.8233

  Columns 19 through 27

    1.1995    1.2359    1.2733    1.3117    1.3512    1.3918    1.4334    1.4762    1.5201
   -0.8269   -0.8314   -0.8367   -0.8428   -0.8499   -0.8579   -0.8668   -0.8767   -0.8877

  Columns 28 through 36

    1.5652    1.6115    1.6590    1.7078    1.7578    1.8091    1.8618    1.9158    1.9713
   -0.8997   -0.9128   -0.9270   -0.9424   -0.9590   -0.9768   -0.9958   -1.0162   -1.0379

  Columns 37 through 41

    2.0281    2.0864    2.1462    2.2075    2.2703
   -1.0609   -1.0854   -1.1113   -1.1387   -1.1676


ifail =

                    0



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Chapter Introduction
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