1. Introduction
Due to the extended use of reconstruction operators in many fields of application, ranging from hyperbolic conservation laws to computer aided geometric design, it is of great importance to dispose of efficient methods to build them for different situations. In general, and for the sake of simplicity, the considered functions are polynomials. High degree polynomials are, however, usually avoided because they are known to generate oscillations and undesirable effects.
Linear operators behave improperly in presence of jump discontinuities, so that different nonlinear operators have emerged to deal with this problematic. Recent approaches to deal with similar problems of functions affected by discontinuities can be found for example in [
1,
2,
3,
4,
5]. And these nonlinear methods also give rise to interesting applications. To mention some of them one can refer to [
6,
7,
8,
9,
10,
11].
In this article we pay attention to one of these operators that was defined in [
12] under the name PPH (Piecewise Polynomial Harmonic). This operator can be seen as a nonlinear counterpart of the classical four points piecewise Lagrange interpolation. The theoretical analysis as much as the practical applications were developed in uniform grids in previous articles (see, for example, [
12,
13,
14,
15,
16,
17,
18]). In turn these reconstruction operators are the heart of the definition of associated subdivision and multiresolution schemes [
5,
19,
20,
21].
In this paper, we extend the definition of the PPH reconstruction operator to data over non uniform grids and we study some properties of this operator. In particular, we analyze the behavior of the operator in presence of jump discontinuities. We prove adaption to the jump discontinuity in the sense that some order of approximation is maintained in the area close to the discontinuity, on the contrary to what happens with linear operators that lose completely the approximation order. We also prove, as much theoretically as in numerical experiments, the absence of any Gibss phenomena.
The paper is organized as follows—in
Section 2 we remind the nonlinear PPH reconstruction operator [
22] on nonuniform grids.
Section 3 is dedicated to study the adaption of the operator to the presence of jump discontinuities, making some emphasis in the order of approximation. In
Section 4 we analyze the behavior of the operator with respect to the Gibbs phenomena. In
Section 5 we present some numerical tests. Finally, some conclusions are given in
Section 6.
2. A Nonlinear PPH Reconstruction Operator on Non Uniform Grids
In this section we recall the definition of the nonlinear PPH reconstruction operator on nonuniform grids, see [
22]. We include the necessary elements for the rest of the article. In [
22] the reconstruction operator is designed to deal with strictly convex functions, albeit it is also of interest in the case of working with piecewise smooth functions affected by isolated jump discontinuities. This will be our case of interest in this section and in the rest of the article.
Let us define a nonuniform grid
in
Let us also denote
the nonuniform spacing between abscissae. We consider underlying piecewise continuous functions
with at most a finite set of isolated corner or jump discontinuities, and let us call
the ordinates corresponding to the point values of the function at the given abscissae. We also introduce the following notations. In first place, the second order divided differences
in second place a weighted arithmetic mean of
and
defined as
with the weights
Given these ingredients in [
22] we can find the following definitions, and results that we will use later.
Lemma 1. Let us consider the set of ordinates for some at the abscissae of a nonuniform grid . Then the values and at the extremes can be expressed aswith the constants given by Definition 1. Given and such that and we denote as the function Lemma 2. If and the harmonic mean is bounded as follows Next definition, which is commonly used in numerical analysis, is going to be essential through the rest of the article.
Definition 2. An expression means that there exist and such that Lemma 3. Let a fixed positive real number, and let and If , and then the weighted harmonic mean is also close to the weighted arithmetic mean Definition 3 (PPH reconstruction). Let be a nonuniform mesh. Let a sequence in Let and be the second order divided differences, and for each let us consider the modified values built according to the following rule
Case 1: If Case 2: If where are given in (5) and with the weighted harmonic mean defined in (6) with the weights and in (3). We define the PPH nonlinear reconstruction operator aswhere is the unique interpolation polynomial which satisfies
According to Definition 3, it is possible to establish a parallelism with Lagrange interpolation, in fact we can write the PPH reconstruction as
where the the coefficients
are calculated by imposing conditions (
12). We explain each one of the two possible local cases, Case 1 or Case 2. The coefficients will have symmetrical expressions.
Case 1. , which means that a potential singularity may lay in
. It has been proposed to replace
with
in Equation (
9) by changing the weighted arithmetic mean in Equation (4b) for its corresponding weighted harmonic mean. This replacement has been performed to carry out a witty modification of the value
in such a way that its difference with respect to the original
is large in presence of a discontinuity, but remains sufficiently small in smooth areas maintaining the approximation order. Lemma 2 is crucial for the adaption in case of dealing with the presence of a jump discontinuity, while Lemma 3 plays a fundamental part in proving fourth approximation order for smooth areas of an underlying function.
In this case the coefficients
of the PPH polynomial read
For our purposes, in the next sections we need to examine deeper the relation with Lagrange interpolation. In particular we get that
and considering the Lagrange interpolation polynomial written in the same form as in (
13), that is
we get that the difference of these coefficients with the ones of
is given by
Case 2. , which means that a possible singularity lies in
. In this case, in Definition 3, the value
is replaced with
by using expression (
10). Similar comments apply in this case due to symmetry considerations. The coefficients for the polynomial (
13) now read
The expressions relating the coefficients of the PPH polynomial with the Lagrange interpolation polynomial now write
In next section, we will study the approximation order of the PPH reconstruction operator in presence of isolated jump discontinuities.
3. Approximation Order around Jump Discontinuities
We are going to study the approximation order of the given reconstruction for functions of class with an isolated jump discontinuity at a given point We consider only the case of working with quasi-uniform grids, according with the following definition.
Definition 4. A nonuniform mesh is said to be a σ quasi-uniform mesh if there exist and a finite constant σ such that
In what follows we give a proposition proving full order accuracy for convex regions of the function, that is fourth order accuracy, and observing that the approximation order is reduced to second order close to the singularities and to third order close to inflection points. We would like to focuss especial attention to the intervals around the discontinuity where the order is reduced, but not completely lost.
Theorem 1. Let be a function of class with a jump discontinuity at the point Let be a σ quasi-uniform mesh in with and the sequence of point values of the function Let us consider such that a fixed positive real number, Ω
the set of all inflexion points of and the distance function defined by Then, the reconstruction satisfies
- 1.
In if and then - 2.
In if and or then - 3.
In if - 4.
In where
Proof. We do the proof point by point.
- 1.
Given the reconstruction operator is built as
From Equations (
2) and (
6) we can write
From hypothesis we have that the initial data are strictly convex in the considered area
(for a concave function the arguments remain the same) and therefore they satisfy
for some
Since second order divided differences amount to second derivatives at an intermediate point divided by two, i.e.,
with
and
Therefore, we have
and from (
21) we get that
Plugging this information into (
17) if
or into (
20) if
we get that
Thus
where
is the Lagrange interpolatory polynomial. Taking into account again the triangular inequality
using that Lagrange interpolation also attains fourth order accuracy.
- 2.
We now prove Point 2. Since
or
and depending on the exact distance to the inflection point we encounter
with
Then from Equation (
21) we directly get
and the rest of the proof follows exactly the same track as in Point 1, giving the enunciated result.
- 3.
For proving Point 3, we observe that in this case
and again following the same track as in previous points we get
and therefore in this case the accuracy is reduced to third order.
- 4.
In order to prove Point 4, let us suppose without lost of generalization that The other case it is proven analogously. Since by hypothesis the function is smooth in , and it presents a jump discontinuity at the interval we have and Therefore .
Let
be the second degree Lagrange interpolatory polynomial built using the three pairs of values
.
where
The difference between these coefficients and the ones of
shown in Equation (
14) is given by
At this stage we distinguish two cases:
Taking into account Equations (
6), (
7) and (
25) and the triangular inequality we obtain
Equations (
6) and (
25) and the triangular inequality lead us to
And these last chains of equations finish the proof.
□
We observe that close to the jump discontinuity, that is, in the intervals and we do not lose all accuracy, but we maintain at least second order accuracy. Unfortunately, in the central interval containing the singularity this approach does not allow us to obtain any gain with respect to other reconstruction operators.
Remark 1. Notice that linear reconstruction operators based on an stencil of four points typically lose the approximation order in three intervals around discontinuities, while the introduced nonlinear reconstruction operator only loses completely the aproximation order in the interval containing the jump discontinuity and maintains at least second order accuracy, that is, in the adjacent intervals. In the interval containing the jump discontinuity the approximation order is lost also in the nonlinear reconstruction strategy, since with point values of the function it is impossible to detect the exact position of the jump discontinuity.
Remark 2. The order reduction due to inflection points can be tackled using a translation strategy in the definition of the Harmonic mean, to avoid arguments of different signs. This strategy complicates the definition of the operator, but it has been satisfactorily introduced on various occasions [12,23]. In practice the translation is needed not only at the interval containing the inflection point, but also in adjacent intervals. 4. Analysis of Gibbs Phenomena around Jump Discontinuities
In this section we are going to give a result analyzing the behavior of the proposed nonlinear reconstruction with respect to the generation of possible Gibbs effects due to the presence of jump discontinuities in the underlying function. In particular we prove the following proposition.
Before enunciating the theorem we introduce some definitions.
Definition 5. Given a σ quasi-uniform grid in we define, for (the larger the k the larger the resolution), the set of nested grids given by where and
Let us also denote the size of a jump discontinuity, the straight line joining the points and the vertical distance from the reconstruction to the horizontal line passing through the middle point of and The respective expressions come given by
We will also use as the maximum distance between and measured perpendicularly to
Theorem 2. Let be a set of nested σ quasi-uniform grids in Let be a function with four continuous derivatives in all the real line with an isolated jump discontinuity at the abscissa μ located at a certain for each where j depends on Then, the reconstruction associated to the data does not generate Gibbs phenomena. In particular, the following statements hold:
- 1.
- 2.
- 3.
- 4.
lies inside the rectangle
- 5.
where
Proof. Let us consider
large enough,
such that
Then
and from (
28), (
29) and (
6) we get
We carry out the rest of the proof addressing point after point.
1. Since only three intervals are affected by the jump discontinuity for construction, then
2. We are going to show now that the oscillations due to the presence of the discontinuity diminish at the interval with k increasing.
In
the PPH reconstruction amounts to
As
the coefficients are given by (
14) adapted to the interval
Taking into account property (
7) of the harmonic mean, we can write
Considering (
31), the distance
can be bounded by
where
3. In applying arguments based on symmetry and taking into account that we also get that
4. In
as
due to (
30), the expression of
according to (
13), (
14), (
18) will be
At this point we consider two subcases depending on and
4.1
The maximum value of the function
in the interval
is either at the extremes of the interval or among any possible critical point
verifying
At the extremes of the interval we have
and the condition is satisfied. We are going to prove that the local reconstruction
lies inside the rectangle
since any critical point
of the function
falls outside the interval
For this purpose, we shall prove that
We start computing
and
Last equations show that is symmetric respect to the vertical axis passing through where it reaches a local maximum since
Evaluating (
35) at
and
we obtain
From (
37) and (
30) we get
To analyze the sign of
we replace in (
36)
by its expression (
28)
and we consider two subcases depending on the sign of
4.1.1 From (
26),
and we get
4.1.2 Again from (
26),
and we get
In both subcases
which together with expression (
38) allow us to write
and therefore
what amounts to say that there is not local maximum value of
inside the interval.
4.2
Following a similar path to case 4.1 we arrive to
and therefore remains inside the rectangle
5. We start computing the points where the slope of the tangent of equals to the slope of the straight line We consider two subcases,
5.1
In this case, the above mentioned points where the tangent of
is parallel to
are given by:
The largest distance from these points to is the maximum distance between and measured perpendicularly to
For both points this distance coincides with
5.2.
The required points
and
in this case take the form:
and
is given by
□
Remark 3. The hypothesis in Theorem 2 concerning the use of a nested set of σ quasi-uniform grids amounts in practice to build the reconstruction with a small enough maximum grid size.
In the next section we carry out some numerical experiments to check that the practical observations coincide with the theoretical results.
5. Numerical Experiment
In this section we present a simple numerical test to validate the theoretical results. Our experiment computes the approximation order of the considered reconstruction in several areas corresponding with the different points in Theorem 1. In particular we measure the approximation order in the following areas, identified with the given acronyms:
- :
In the subinterval containing the discontinuity.
- :
In a region where the function is smooth without inflexion points.
- :
In a region where the function is smooth but contains a inflexion point.
- :
In a region close to the inflexion point without containing it.
- :
In the subinterval just to the right of the one containing the singularity.
Let
be a non uniform grid in
and
the following smooth function with a jump discontinuity at
and an inflexion point at
Given the initial abscissas
we consider the set of nested grids
where
and
with
For each level of resolution
k we build the PPH reconstruction using the data
computing the approximation errors in infinity norm with respect to the original function using a denser set of abscissas, that is, we compute a numerical approximation of
Then, we compute the numerical approximation order as
Notice that due to Theorem 1 we can assume that for fine enough grids
In
Table 1 and
Table 2 we present the errors committed by Lagrange and PPH reconstructions respectively when using as initial nodes the defined nested grids
The errors appear separately for each kind of region
,
,
,
and
The largest error comes near the jump discontinuity for Lagrange reconstruction, as it can be observed in the column corresponding with
.
In
Table 3 we present the obtained approximation orders for the studied PPH reconstruction and just for the sake of comparison we also add the approximation orders for the classical four points piecewise Lagrange polynomial interpolation. We have computed the approximation order in the specified different regions
,
,
,
and
More in concrete, in the case of region
we use the interval
for the x variable, in the case of region
the interval
and in the case of region
the intervals
, where
k indicates the resolution level and the index
is such that the inflexion point falls into the interval
for each
We can observe that in the region
both reconstructions are affected by the jump discontinuity and they lose the approximation order due mainly to the subinterval containing the discontinuity. In the region of type
both reconstructions attain fourth order accuracy as expected. In the case
the PPH reconstruction reduces the approximation order to third order due to the presence of the inflexion point. Similarly in the vicinity of the inflexion point, region
the PPH reconstruction stays between
and
In the adjacent intervals to the singularity, case
we clearly observe an improvement with respect to Lagrange interpolation, since we obtain order
while Lagrange completely loses the approximation order. Notice that the order reduction produced in the regions
and
occurs in very limited areas and it can be corrected using a translation strategy (see [
12,
23]) that we have not implemented in this experiment with the aim of studying the original reconstruction operator.
In
Figure 1 we plot the function
and the Lagrange and PPH reconstructions obtained from the initial grids
We can see that around the singularity, Lagrange reconstruction looses the approximation order and the Gibss phenomena appears. In this zone, PPH reconstruction performs in a more proper way, avoiding any Gibbs effects. We can see that no oscillations appear in the PPH reconstruction even for the coarsest grid. These observations can be seen more clearly in
Figure 2 where we have plotted a zoom of this region for
for both reconstruction operators Lagrange and
We also point out that the oscillations due to the jump discontinuity in Lagrange reconstruction do not diminish to zero with the subdivision level. In fact, from
we have check out that the reconstruction values at the local maxima and minima of the oscillations remain almost constant.
In the jump interval the distance
decreases as
increases, since
In
Table 4 the values for
are shown. We can see that at a certain subdivision level the given values are approximately decreasing with the ratio
Therefore, PPH reconstruction approaches to the straight line
as
increases.
6. Conclusions
We have studied the behavior of the PPH reconstruction operator in presence of jump discontinuities for the case of working with quasi-uniform grids. For this purpose, the arithmetic and harmonic means used in the uniform case are changed for weighted means with concrete weights, so that the main properties that allow for maintaining order of approximation in smooth areas and adaptation near singularities continue being true.
A explicit result concerning the approximation order, Theorem 1, has been proved, showing at least second order of approximation for the adjacent intervals to the one containing the jump discontinuity, and ensuring fourth order of approximation in convex (concave) parts of the function far from inflexion points. At a interval containing a inflexion point we get third order of approximation and in the vicinity the order grows progressively till fourth order.
A main result of this article is Theorem 2 in
Section 4 proving that the presented reconstruction operator does not generate any Gibbs phenomena in the concrete sense indicated in the enunciate for
quasi-uniform grids where the maximum space between nodes of the grid is small enough.
Finally we have carried out some numerical experiments to reinforce the theoretical results proven as much in Proposition 1 as in Theorem 1.