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# LSodar

*LSodar* (short for Livermore Solver for Ordinary Differential equations, with Automatic method switching for stiff and nonstiff problems, and with Root-finding) is a numerical solver providing an efficient and stable method to solve Ordinary Differential Equations (ODEs) Initial Value Problems.

### Description

Called by xcos, *LSodar* (short for Livermore Solver for Ordinary Differential equations, with Automatic method switching for stiff and nonstiff problems, and with Root-finding) is a numerical solver providing an efficient and stable variable-size step method to solve Initial Value Problems of the form:

LSodar is similar to *CVode* in many ways:

- It uses variable-size steps,
- It can potentially use
*BDF*and*Adams*integration methods, *BDF*and*Adams*being implicit stable methods,*LSodar*is suitable for stiff and nonstiff problems,- They both look for roots over the integration interval.

The main difference though is that *LSodar* is **fully automated**, and chooses between *BDF* and *Adams* itself, by checking for stiffness at every step.

If the step is considered stiff, then *BDF* (with max order set to 5) is used and the Modified Newton method *'Chord'* iteration is selected.

Otherwise, the program uses *Adams* integration (with max order set to 12) and *Functional* iterations.

The stiffness detection is done by step size attempts with both methods.

First, if we are in *Adams* mode and the order is greater than 5, then we assume the problem is nonstiff and proceed with *Adams*.

The first twenty steps use *Adams / Functional* method.
Then *LSodar* computes the ideal step size of both methods. If the step size advantage is at least *ratio = 5*, then the current method switches (*Adams / Functional* to *BDF / Chord Newton* or vice versa).

After every switch, *LSodar* takes twenty steps, then starts comparing the step sizes at every step.

Such strategy induces a minor overhead computational cost if the problem stiffness is known, but is very effective on problems that require differentiate precision. For instance, discontinuities-sensitive problems.

Concerning precision, the two integration/iteration methods being close to *CVode*'s, the results are very similar.

### Examples

The integral block returns its continuous state, we can evaluate it with *LSodar* by running the example:

// Import the diagram and set the ending time loadScicos(); loadXcosLibs(); importXcosDiagram("SCI/modules/xcos/examples/solvers/ODE_Example.zcos"); scs_m.props.tf = 5000; // Select the LSodar solver scs_m.props.tol(6) = 0; // Start the timer, launch the simulation and display time tic(); try xcos_simulate(scs_m, 4); catch disp(lasterror()); end t = toc(); disp(t, "Time for LSodar:");

The Scilab console displays:

Time for LSodar: 9.546

Now, in the following script, we compare the time difference between *LSodar* and *CVode* by running the example with the five solvers in turn:
Open the script

Time for LSodar: 6.115 Time for BDF / Newton: 18.894 Time for BDF / Functional: 18.382 Time for Adams / Newton: 10.368 Time for Adams / Functional: 9.815

These results show that on a nonstiff problem, for the same precision required, *LSodar* is significantly faster. Other tests prove the proximity of the results. Indeed, we find that the solution difference order between *LSodar* and *CVode* is close to the order of the highest tolerance (
*y _{lsodar} - y_{cvode}*
≈

*max(reltol, abstol)*).

Variable-size step ODE solvers are not appropriate for deterministic real-time applications because the computational overhead of taking a time step varies over the course of an application.

### See also

### Bibliography

ACM SIGNUM Newsletter, Volume 15, Issue 4, December 1980, Pages 10-11 LSode - LSodi

### History

Version | Description |

5.4.1 | LSodar solver added |

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