In the case of a linear problem, where f ( q ) = a q, and without loss of generality, we'll assume that a > 0, the upwinded Godunov method yields:
22.
One approach is to use a linear approximation to the nonlinear model, which may be a close approximation in the vicinity of the solution, and then apply the analysis for a linear problem to find an approximate confidence region.
23.
In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving ? " t " gets mapped into an infinite-dimensional linear problem involving " U ".
24.
The second is a matter of geometry, on the model of parallel lines; it is something that can be discussed for linear problems by methods of linear algebra, and for non-linear problems in projective space, over the complex number field.
25.
The second is a matter of geometry, on the model of parallel lines; it is something that can be discussed for linear problems by methods of linear algebra, and for non-linear problems in projective space, over the complex number field.
26.
However, iterative methods are often useful even for linear problems involving a large number of variables ( sometimes of the order of millions ), where direct methods would be prohibitively expensive ( and in some cases impossible ) even with the best available computing power.
27.
The voltage-control model requires an exponential function to be taken into account, but when it is linearized such that the transistor can be modeled as a transconductance, as in the Ebers Moll model, design for circuits such as differential amplifiers again becomes a mostly linear problem, so the voltage-control view is often preferred.
28.
For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM ( the quantities C _ { D }, S _ { D } and W _ { D }, Non-linear models for which such convergence proof of the GDM have been carried out comprise : the Stefan model of a melting material, two-phase flows in porous media, the Richards equation of underground water flow, the fully non-linear Leray Lions equations.
29.
The book starts with a more general theory for all relevant flow and transport phenomena on the basis of the continuum mechanics, systematically develops the basic framework for important classes of problems ( e . g ., multiphase / multispecies non-isothermal flow and transport phenomena, variably saturated porous media, free-surface groundwater flow, aquifer-averaged equations, discrete feature elements ), introduces finite element methods for solving the basic multidimensional balance equations, in detail discusses advanced numerical algorithms for the resulting nonlinear and linear problems, and completes with a number of benchmarks, applications and exercises to illustrate the different types of flow, mass and heat transport problems ( e . g ., subsurface flow and seepage problems, unsaturated-saturated flow, advective-diffusion transport, saltwater intrusion, geothermal and thermohaline flow ).
