Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations

ly u t is nothing more than an element in a Banach space B and this leaves room for applications to a wide variety of problems in di erential equations As in the examples above B might be L and A might be x or x and homogeneous boundary conditions could be included by restricting B appropriately More generally u t could be an vector valued function of multiple space variables The next step is to de ne a general nite di erence formula In the abstract setting this is a family of bounded linear operators Sk B B THE LAX EQUIVALENCE THEOREM TREFETHEN where the subscript k indicates that the coe cients of the nite di erence formula depend on the time step We advance from one step to the next by a single application of Sk v Skv n hence v S k v where S k abbreviates Sk n Watch out for the usual confusion of notation the n in v is a superscript while in S k it is an exponent For simplicity but no more essential reason we are assuming that the problem and hence Sk have no explicit dependence on t But Sk does potentially depend on k and this is an important point On the other hand it does not explicitly depend on the space step h for we adopt the following rule h is a xed function h k of k For example we might have h k constant or h q k constant If there are several space dimensions each may have its own function hj k More generally what we really need is grid k not h k ! there is no need at all for the grid to be regular in the space dimensions EXAMPLE Lower order terms For the UW model of ut ux the discrete solution operator is de ned by Skv n j v n j v n j v n j and if is held constant as k this formula happens to be independent of k The natural extension of UW to ut ux u on the other hand is Skv n j v n j v n j v n j kv n j and here there is an explicit dependence on k This kind of k dependence appears whenever the operator A involves derivatives of di erent orders Implicit or multistep nite di erence formulas are not excluded by this for mulation As explained in x an implicit formula may still de ne a bounded operator Sk on an appropriate space such as h and a multistep formula can be reduced to an equivalent one step formula by the introduction of a vector w v v s Let us now be a bit more systematic in summarizing how the setup for the Lax Equivalence Theorem does or does not handle the various complications that make real problems di er from ut ux and ut uxx THE LAX EQUIVALENCE THEOREM TREFETHEN Nonlinearity The restriction here is essential the Lax Richtmyer theory does not han dle nonlinear problems However see various more recent papers by Sanz Serna and others Multiple space dimensions Implicit nite di erence formulas Both of these are included as part of the standard formulation Time varying coe cients Initial value problems with time varying coe cients are not covered in the description given here or in Richtmyer and Morton but this restriction is not essential The theory can be straightforwardly extended to such problems Boundary conditions Space varying coe cients Lower order terms All of these are included as part of the standard formulation and they have in common the property that they all lead to nite di erence ap proximations Sk that depend on k as illustrated in Example above Systems of equations Higher order initial value problems Multistep nite di erence formulas These are covered by the theory if we make use of the usual device of reducing a one step vector nite di erence approximation to a rst order initial value problem As in Chapter we begin a statement of the Lax Richtmyer theory by de ning the order of accuracy and consistency of a nite di erence formula fSkg has order of accuracy p if ku t k Sku t k O k p as k for any t T where u t is any su ciently smooth solution to the initial value problem It is consistent if it has order of accuracy p There are di erences between this de nition and the de nition of order of ac curacy for linear multistep formulas in x Here the nite di erence formula is applied not to an arbitrary function u but to a solution of the initial value THE LAX EQUIVALENCE THEOREM TREFETHEN problem In practice however one still calculates order of accuracy by substi tuting formal Taylor expansions and determining up to what order the terms cancel Exercise Another di erence is that in the case of linear multistep formulas for ordinary di erential equations the order of accuracy was always an integer and so consistency amounted to p Here non integral orders of accuracy are possible although they are uncommon in practice EXAMPLE Non integral orders of accuracy The nite di erence approximation to ut ux Skv n j v n j v n j v n j k p constant is a contrived example with order of accuracy p if p is any constant in the range A slightly less contrived example with order of accuracy p is Skv n j v n j k h v j v n j with h k p for any p As with ordinary di erential equations a nite di erence formula for a partial di erential equation is de ned to be convergent if and only if it con verges to the correct solution as k for arbitrary initial data fSkg is convergent if lim k nk t kS ku u t k for any t T where u t is the solution to the initial value problem for any initial data u Note that there is a big change in this de nition from the de nition of convergence for linear multistep formulas in x There a xed formula had to apply successfully to any di erential equation and initial data Here the di erential equation is xed and only the initial data vary The de nition of stability is slightly changed from the ordinary di erential equation case because of the dependence on k fSkg is stable if for some C kS k k C for all n and k such that nk T THE LAX EQUIVALENCE THEOREM TREFETHEN This bound on the operator norms kS k k is equivalent to kvk kS k v k Ckv k for all v B and nk T Here is the Lax Equivalence Theorem compare Theorem LAX EQUIVALENCE THEOREM Theorem Let fSkg be a consistent approximation to a well posed linear initial value problem Then fSkg is convergent if and only if it is stable Proof Not yet written The following analog to Theorem establishes that stable discrete for mulas have the expected rate of convergence