I see this in so many maps. Normally, the water flows downhill because of gravity but here it is the opposite, rivers will go above the hills. When you mention this incoherence, the authors refuse to change it because that's how they want it. Apparently, the laws of physic does not need to apply with rivers.
Is there any phenomenon that could explain this even if it's only on a short distance?
Since the question specifically refers to maps, I should mention that it is quite normal for it to appear that rivers run towards higher ground, such as hills or mountains, on a map showing only the large scale details. This happens because you can't see the actual terrain the river flows through, which can be much lower than the large scale features that you can see on the map. The river can erode itself quite a deep and narrow canyon given time.
Two possibilities are obvious:
While this does not quite answer the question as asked, I think there is some value in directly addressing the stated motivation for the question.
Water can flow uphill in a formation known as a hydraulic jump. These phenomena are most visible in two types of locations:
In some rapids, the speed of a river drastically decreases when fast-moving water discharges into a slow region of the river. If the speed of the water is greater than a certain threshold (see the linked article above for calculation of this speed), the force that the fast water exerts as it slows down can force the following water to rise slightly.
Some dam spillways use an engineered hydraulic jump to slow the speed of the water coming off the spillway. These represent a best-case scenario: the spillway is a smooth, low-friction surface, and it usually is a long, steep fall with the jump directly at the bottom (no horizontal distance for energy to be lost). However, even under ideal conditions the jump will be tiny compared to the initial fall. Even a "ski jump" type spillway, where the water is launched into the air, achieves very little height.
Finally, as Serban alluded to in a (now deleted) comment, if the speed of the water is high enough to force it uphill over any significant distance, it will tend to erode the obstacle over time, or push out of the riverbed and flow around the obstacle. (See the answers to this question for some of the problems associated with trying to naturally constrain high-pressure water.)
Here's one river attempting to flow vertically downhill, and failing.
All you need is a good stiff breeze (in Skye terms) or perhaps a full gale in normal language, and you can induce water to flow uphill, at least on a local scale.
You will note that the shape of the beach is fairly close to the curve of a turbine blade, to accelerate onshore winds into an upwards direction. Not so apparent is the way the curve of the bay helps concentrate the wind onto that spot.
(Location: Talisker Bay, Isle of Skye. Windspeed : moderate, probably 40mph or less)
With the science covered, I thought I'd mention two examples on Earth of this occurring for relatively longer times and distances.
The first is the Tonle Sap in Cambodia, which is a tributary of the Mekong. It reverses directions twice a year in response to seasonal rainfall changes. In the wet season, the larger Mekong river starts flooding. These flood waters then push up the Tonle Sap. Essentially, the regular downhill flow is overcome by uphill flow caused by the large increase in the volume of water.
A second example, which is cheating a little, are the sub-glacial rivers of Antarctica. Here gravity is less dominant in determining direction of flow, as the pressure of the ice takes over. I have not yet been able to find a reference for the distances that may be involved here, and it appears to be an active area of research.
As user3453518 says, this could happen for a short distance by inertia. I advise you to read his answer; there are some very good points made well in it. I'll add some science to it.
The formula for gravitational potential energy is
$$ \large E_\text{GP} = mgh $$
where $m$ is the mass of the object in question, $g$ is the gravitational constant, and $h$ is its height above the turning point. On Earth, $g = 9.81\text{ Nkg}^{-1}$.
So let's say we have 1 cubic metre of pure water, weighing in at 1000kg. It's starting down a hill of 50m vertical height (we'll assume this hill is somehow made of super-hydrophobic material so as to easily discount friction). It has:
$$ E_\text{GP} = 1000 \times 9.81 \times 50 $$ $$ E_\text{GP} = 490.5\text{ kJ} $$
At the bottom of this slope, since energy is conserved and we're discounting friction, it will also have 490.5 kJ. It'll be travelling at a velocity we can determine using the formula for kinetic energy:
$$ E_\text{K} = \frac{1}{2} mv^2 $$ $$ v = \sqrt{\frac{2E_\text{K}}{m}} $$ $$ v = 31.32\text{ ms}^{-1} $$ or about 70 mph.
Using this, we can show how high it could go if projected straight up with no loss of kinetic energy:
$$ v_f^2 - v_i^2 = 2as $$
We're looking for $s$, the displacement from the original position. $a$ is deceleration due to gravity, so just −9.81, $v_f$ is the final velocity (0) and $v_i$ the initial velocity which we just found.
$$ s = \frac{v_f^2 - v_i^2}{2a} $$ $$ s = \frac{0 - 980.9424}{2\times -9.81} $$ $$ s = 49.9971\text{m} $$
That's close to the original height of 50m (it's not exact because of my rounding)—but here's the crucial part: it's not over it. Unless you can manipulate gravity specifically for water without affecting anything else on the planet, it's physically and scientifically impossible to go over the value of the original height. Moreover, in these calculations I have excluded several important considerations such as friction and air resistance. In reality, you'll never get anywhere near this value.
For a very short span, this can be explained by inertia. But, this means that the river might climb a slope no bigger than what it came down.
IE.: The river came down a 10m steep descent, followed by a very short plain (to avoid losing speed to friction) followed by, say, a 3m climb.
But, if you measure the place where water started versus where the water ends, the start level MUST be higher than the ending level, because all of the water's kinetic energy comes from gravitational potential energy. Anything else would defy the laws of physics. Gravity is a fundamental law of the universe and one where this does not hold true would make for a totally different universe. If gravity could be defied in such way, you could very well have a river that does not follow a riverbed. Water might take off right from the sea, climb to stratosphere and flee the planet, etc. To remove gravity as a limitation means to remove it as a limitation everywhere.
To add to the other answers, there's also the possibility of the water appearing to run uphill due to a an optical illusion.
I'm not aware of any river where this is the case, but since it happens with roads I see no reason why it couldn't also happen with rivers.
For particularly extreme topology, a possible explanation could be vertical deflection due to gravitational anomalies. Basically, the local gravity at a point on the surface of a planet may not point directly towards the planet's center of mass, in a similar way to how a moon does. The water is still flowing downward - the difference is that that might not be the same as decreasing elevation.
However, this effect is very small - the largest vertical deflection found on Earth (in some parts of the Himalayas) is a measly 100 arc seconds, or about 5 inches per thousand feet (or 48 cm per km). So, assuming that you have a super-massive mountain next to a super-flat plain, you might have a river that flows in a direction of increasing elevation continuously in a small area, for a few decades until it eroded the land enough to change the direction again.
I just skimmed over the other answers but I think they all miss the point of water actually flowing uphil over an extended distance. Actually this is possible! There's a road (IIRC) "devil's road" in North Africa. This road runs up a lonely hill in otherwise flat terrain. If you put a ball on this road, it will start rolling uphill slowly. People thought devils would be doing that, hence the name.
In fact the hill is just the peak of a massive stone formation, which due to its increased density "bends" the gravitational field of earth a little. The effect is just enough to overcompensate the slope of the hill making balls and potentially also water run uphill. A pendulum would point slightly sideways compared to the flat terrain. So it's more a visual thing: If you define "down" perpendicular to the majority of ground you can see, this is what you're searching for. However if "down" is defined by a pendulum, the water is still flowing downhill up that hill.
Extending on that a shallow river might seem to run uphill, if some extremely big and dense mass is under the hill, which in turn is surrounded by flat and less dense terrain, e.g. barren soil. It's important to define where the river runs to. I'd imagine a small lake from, which the water evaporates quickly enough to make space for more water to run uphill.
The above makes for a shallow and slow stream under extreme conditions. You can go a little bigger, if you add the wind idea: Constant wind from one direction will make trees grow into that direction, instead of growing straight up. This will add to the visual effect of the landscape and add more push to the water running uphill. The hillside can be or at least look steeper. However if more water ends up in the hill's lake, where does it go? I'd recommend a small siphon draining the lake through an underground river. The pressure of the lake regulates the speed of the underground river so that the lake won't overflow easily. You could have stories about people vanishing when bathing in the hill's lake, because they could be sucked into the underground river.
Actually I just noticed there's already an answer by celtschk mentioning the exact same effect. Strange that my two paragraphs were too few but his two lines are good. Maybe someone can merge them up into one answer and then delete mine, because celtschk came up with it first.
So, working off of @NoAnswer's answer, you could have a bit of ultra-dense (say some neutron star material) formations hidden in the top of hill, which has such extreme gravity that everything flows upward toward it. Not enough to cause a black hole, but enough to overcome the gravity of the entire rest of the planet. And this stuff would have to be sitting on some very strong foundation so as to resist sinking to the core of the planet.
ALTERNATELY You could have some algae or fish spawning goo that is seasonally secreted into the river that turns that section of the river into something like polyethelene oxide ("the liquid that pours itself").
It would work like this: river runs parallel to raised area with sharp drop off, then down hill. Fish spawn temporarily changes the viscosity of river, which then tends to pool up in places in the raised area. Eventually a pool reaches the edge of raised area and spills over, which pulls the down-river part of the river back uphill.
I don't know.
Glacial compression can cause water to flow uphill as the pressure from the ice forces the water to travel upwards - this can be seen in areas of the Pennines and can travel a few hundred metres with ease.
