Numerics Modules

These modules implement various numerical algorithms that are needed for the rest of MESA. The num module has a selection of solvers for ODEs and DAEs. Those solvers in turn use linear algebra routines from the mtx module. In addition there are modules for interpolation in one or two dimensions.

MESA/NUM
MESA/MTX
MESA/INTERP_1D
MESA/INTERP_2D
MESA/MEBDFI_95

MESA/NUM

explicit solvers (for non-stiff problems)
explicit Runge-Kutta ODE integrators of orders 5 and 8
dense output; automatic stepsize control; monitoring for stiffness
implicit solvers (for stiff problems)
linearly implicit Runge-Kutta; 2nd, 3rd, and 4th order versions
implicit extrapolation integrators of variable order (midpoint or euler)
support dense, banded, or sparse matrix routines
analytic or numerical difference jacobian
explicit or implicit ODE systems
dense output; automatic stepsize control
Newton-Raphson solver for multidimensional nonlinear root finding
square or banded matrix
analytic or numerical difference jacobian
when possible, reuses jacobian to improve efficiency
uses line search method to improve convergence
safe 1d root finding
uses alternating bisection and inverse parabolic interpolation
have option to use derivative as accelerator (newton method)

Reference

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics 14, Springer-Verlag, second edition 1996.

MESA/MTX

matrix decompositions and solves
BLAS and LAPACK for dense, banded, and tridiagonal
option for using GotoBLAS or Intel’s MKL
support for sparse matrix decompositions and solves
includes subset of SPARSKIT sparse matrix iterative solver
has interface to SuperLU sparse matrix direct solver

References

Kazushige Goto, GotoBLAS website.

Yousef Saad, SPARSKIT website.

Xiaoye S. Li, SUPER LU website.

MESA/INTERP_1D

monotonicity preserving cubic interpolation
piecewise monotonic cubic interpolation

References

Steffen, M., “A simple method for monotonic interpolation in one dimension”, Astron. Astrophys., (239) 1990, 443-450.

Huynh, H.T., “Accurate Monotone Cubic Interpolation”, SIAM J Numer. Anal. (30) 1993, 57-100.

Suresh, A, and H.T. Huynh, “Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping”, JCP (136) 1997, 83-99.

MESA/INTERP_2D

bivariate interpolation and surface fitting
rectangular-grid or scattered set of data points

References

Hiroshi Akima, Rectangular-grid bivariate interpolation and surface fitting. ACM Algorithm 760., ACM Trans. Math. Software (22) 1996, 357-361.

Robert J. Renka, Cubic Shepard method for bivariate interpolation of scattered data. ACM Algorithm 790., ACM Trans. Math. Software (25) 1999, 70-73.

Doug McCune, PSPLINE Home Page

MESA/MEBDFI_95

MEBDFI_95 is an f95 library for solving stiff initial value problems with fully implicit systems of differential algebraic equations:

g(t,y,y’)=0, with vector y=(y(1),y(2),y(3),.....,y(n)).

The code is based on MEBDFI, backward differentiation formulas (BDF) as modified (M) and extended (E) by Jeff Cash, and then modified some more by him to handle fully implicit equations (I). So the name can be parsed as M-E-BDF-I, a modified extended backward difference formula scheme for stiff fully implicit initial value problems. It has automatic step-size selection and automatic order control up to 7th order.

References

J. R. Cash, Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations”, Proc. Roy. Soc. London, Ser. A, vol 459, (2003) pp 797-815.

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer 1996, Page 267.


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