Implicit vs. Explicit
I would like to know why are impicit schemes in general more stable than explicit schemes.
Re: Implicit vs. Explicit
(1). In general, explicit methods compute the next time step solution directly based on the known current time information. It is therefore necessary to keep the time step within the limit of physics in order to obtain accurate sequential solution in time. (2). If the solution is linear in time, and the rate of change in time is linear or constant, then the time step can be made larger and still have accurate solution. (3). But, if the time rate of change is non-linear, that is, it is a curve in solution vs time space, the direct computation of the solution will always have errors in it. This numerical integration error can be magnified or accumulated in the path of time integration. (4). And at some point in time, it can prevent the further time integration because of the poor quality in the integrated solutions. (5). Actually, when the time rate of change is not constant or linear, the explicity model of integration is not accurate. The solution depends on the time continuously, and not just at the current time step. (6). The implicit methods make use of this property, by evaluating the solutions at two or more time space. In this way, one can improve the accuracy of the solution, in principle. There are many examples in the integration of an ordinary differential equation by using multi-steps. (7). Naturally, the difficulty with the implicit methods is in finding the solution itself. The solutions at non-current time space will be unknown. It is more difficult to solve. (8). This improved solution accuracy associated with the implicit methods (methods involved with more than the current time space information into the future), in principle, will be more accurate. And thus stays bounded within limits. A common word is the solution is more stable, it does not accumulate errors and becomes un-bounded, or diverge. (9). In general, if a method make use of the current time space information and the possible future solutions properties in the formulation, then the improved solution accuracy will keep the solution bounded. (thus more stable) (10). The iterative method is similar to the time integration method, if one looks at the iteration number as if it is the time step variable.
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