This isn't my specialty, but I'm currently finishing my Ph.D. in regularity theory of nonlinear elliptic pde (so I have some knowledge of PDEs:P). I also have written a few numerical solvers for parabolic, elliptic and hyperbolic PDEs. Basically, I agree with what macklin01 said; especially, the statements regarding convergence of characteristics and solvers preserving an entropy criterion.
Really, I believe you need to understand your equation a little more theoretically before trying to write a solver for it. If you wish to solve for situations where shocks form, this will limit your choices quite a bit (both pseudo-spectral and FD methods will be poor choices in this scenario). If you are trying to numerically solve such an equation, you should be able to do some analytical calculations to see whether or not shocks will form. If you are not familiar with these calculations, they are spelled out in Evans book on PDE (and there are plenty of ways to analytically approximate wave equations). The only info I have on solving with shocks essentially deals with dividing up your spatial domain and then approximating your solution by solutions to the Riemann problem on each of these subdivided domains (with BCs correlating to the solution from the previous time step). That was from a talk I heard from a few years ago from so the details are a bit hazy. Nonetheless, the issues with shocks are HEAVILY studied.... for various military purposes.
If you aren't dealing with shocks and have a somewhat regular spatial domain, pseudo-spectral methods (i.e. using Fourier transforms) will give you good results. Just thinking about it heuristically, you should be able to get an idea for an adequate spectral band for your approximation if you can prove some type of gradient bound for analytic solutions to the problem. If there is no gradient bound (i.e. could mean a shock formation), this probably isn't the place to start.
If you are dealing with irregular domains then my guess is to look into FD methods using adaptive grids (i.e. densities would correlate to increasing gradients of numerical approximations). I assume you already know about implicit and explicit FD methods and various interpolations between the two... so I won't belabour that point.
I have no knowledge of using FE methods or of the level set methods that macklin01 mentioned in the setting of hyperbolic equations. Sorry.
You may also want to look into using fractional time steps. I.e. you take a half time-step using one numerical methodology, then you take the second half of the time step using another. Usually, the first half time step gives you a good approximation with less stability, the second half stabilizes the approximation from the first half. This is a very fast and loose explanation. Fractional time steps are well covered in literature (numerical recipes in C even briefly mentions them). Personally, I have found them very useful when using more exotic numerical methods that weren't so stable.
In regards of what to use to write the solver in. Personally, I always have used F90 or perhaps C to code up anything I wrote in this regard; but if you only plan on doing a few runs of the solver to get a few results, then matlab is a good fast alternative to F90. If you plan to run many simulations using your solver, then nothing will beat the speed of a well-coded solver written in F90.
As for visualization, time evolution of select level sets is the most obvious choice (like macklin01 said). It really depends on what you want to convey with your data though (as with any visualization). Thus, if you are trying to see just spatial representations of solutions, level sets are a good representation for 3D wave equations. You could also try color scales corresponding to your solution and looking at various slices of your domain; but this obviously requires some knowledge of where to look in your solution domain for certain phenomena. If you want to see shock formations, you could make 3D density plots based on the gradient value of your numerical app
I am currently a Ph.D. student in mathematics and have been teaching tutorials and courses since my second year of Uni. My feeling about using computers and various pieces of technology to teach either one of these topics is that they are counterproductive. I had two incredible teaches as an undergraduate in math and physics, and NEITHER used technology to make them effective teachers. What made them good was that they knew how to explain seemingly complex ideas in simple intuitive terms; and they actually helped students to visualize mathematics/physics in their head. They essentially taught students how to teach themselves new ideas.
The fact of the matter is, powerpoint style lectures have been shown to harm the attention span of students. It's like when you were in grade school and the teacher put on a movie for class; you zone out almost immediately. As for using computers to aid in 2D and 3D visualization, frankly I don't think this is a particularly good idea either. Students aren't always going to have a nice 3D movie given to them when they are faced with a new concept in math or physics. In higher level mathematics, naive 2D and 3D representations of concepts can be either misleading or incorrect; and making a student dependent on such simple visualizations cripples their ability to visualize more abstract concepts in effective ways. I believe in teaching students how to visualize things in their head; that is, how to sketch out pieces of a problem from something they see in the form of mathematics.
The point is, anyone can sit down and read a textbook... and if you are a teacher that essentially just regurgitates what is inside the textbook, you will never be a great teacher no matter how much technology you use. A great teacher or lecturer is the one who can show the student a path to original intuition. This will enable the student to draw their own visualizations, etc. when confronted with a new problem. This is precisely what made Richard Feynmann such an incredible teacher; he used intuition to make the most complex concepts intuitively clear to anyone... then he reconciled that intuition with mathematics. The mark by which I judge myself in understanding a subject is that I should be able to pull someone off the street and get at least the idea of what I'm talking about across to them in a few minutes. As I deal mostly in differential equations, I obviously can't do this just by writing equations; but it can be done intuitively by using simple explanations, heuristic examples and sketches.
I'm also a firm believer in the idea that students are meant to take notes in class. Writing information down in a note taking fashion forces the student to acknowledge the content being taught on some level. If nothing else, it reinforces what the lecture is saying. I know taking notes from a lectures that just dumps out information is difficult; but I believe them to be remarkably effective when paired with a teacher who conveys more in the way of intuition than statements of fact. The textbook is there to give the data... as a teacher you are meant to give the intuition to the student so they can sit down and understand that data. Forcing them to write down notes on their own will enable them to build a more solid connection with the intuition you are (hopefully) conveying.
In the end, learning by rote is the lowest form of understanding. Next comes the ability to do problems that you are already acquainted with. After this level comes the ability to solve slightly new problems by combining methods you used to solve previous problems. After this level, one gets into various levels of intuition which enable one to solve completely new problems. I try to teach my students in the hopes that they are seeking at least this level of understanding. I wish more teachers would spend more time working on their lectures along these lines as opposed to spending so much time and money to make their lectures nothing more than a powerpoint presentation.
I can't believe that couples actually argue over such trivial crap. My wife and I have been nigh unseperable since before we were married (almost a year now). Yes, our marriage is still fairly new; but we both have previous experience with relationships that have lasted over the course of years. Since being married we've had to deal with several situations that I believe would strain the strongest of marriages. First, I moved to Australia a few years ago from the US; bringing my wife (also an American) to live with me here was no easy task. Issues and costs associated with this move alone were difficult for a low-income couple like us. We've had to deal with issues regarding families, money, jobs, housing, and even education (I'm finishing post-grad studies) since being married; and we've not had the altercations listed in this article or the slashdot posts.
Sharing a blog, emails, Tivo?! Who the hell really cares about this type of thing? My wife and I have separate emails just because it's confusing if we combine them. My wife knows all my passwords and I hers; so we can access each other's computers, accounts, whatever if we really need to. We help each other with our computers as well... she's an interface designer and I like playing around with hardware, kernels etc. It works out really nicely (we both are Arch users). We both have almost identical tastes in music, books and movies; this was one of the biggest reasons we hit it off so well in the beginning. Now I admit, I lucked out having a wife who actually likes the movie "Doom" and is a huge Tool fan; but I would have thought a common interest in these things would be had by many married couples. While we both like videogames, the only TV shows we really care to watch are Southpark and BSG. Again, I know I'm lucky; but then again, I didn't choose to marry someone with whom I really didn't have that much in common.
What I really think is the bigger issue here is porn. I, like every guy here, had a big porn collection before meeting my now wife. Being single, I was perfectly happy wacking off to porn whenever I felt like it; and I still don't see a problem with this. That being said, whenever I started dating a woman in the past; I would stop looking at porn on my own. I look at it like this, I would not be happy to walk in and see someone I'm dating masturbating to internet porn; so I don't do it. I have never been asked to stop looking at porn by anyone I've dated; it's just something I did on my own. In my view, masturbating to porn is the most effective way to make your partner feel utterly unattractive, and it will build resentment. I really feel that it's this building resentment that fuels arguments over trivialities; people don't want to take on the nasty issue, so they just berate each other over trivialities. There are many reasons why I really like my wife; but, one big one, is that are views on sexuality are the same. We both genuinely like sex as both a physical and mental act; and we don't deny each other's physical or mental desires. I believe this is healthy and unfortunately, somewhat rare given how twisted our societies' view on sex is. As for having sex with the same partner getting old; well, I don't really experience that. People's minds and bodies are infinitely fascinating; and so, I haven't ever felt boredom with a sexual partner even if we've been together for years.
Slashdot Mirror
This isn't my specialty, but I'm currently finishing my Ph.D. in regularity theory of nonlinear elliptic pde (so I have some knowledge of PDEs :P). I also have written a few numerical solvers for parabolic, elliptic and hyperbolic PDEs. Basically, I agree with what macklin01 said; especially, the statements regarding convergence of characteristics and solvers preserving an entropy criterion.
Really, I believe you need to understand your equation a little more theoretically before trying to write a solver for it. If you wish to solve for situations where shocks form, this will limit your choices quite a bit (both pseudo-spectral and FD methods will be poor choices in this scenario). If you are trying to numerically solve such an equation, you should be able to do some analytical calculations to see whether or not shocks will form. If you are not familiar with these calculations, they are spelled out in Evans book on PDE (and there are plenty of ways to analytically approximate wave equations). The only info I have on solving with shocks essentially deals with dividing up your spatial domain and then approximating your solution by solutions to the Riemann problem on each of these subdivided domains (with BCs correlating to the solution from the previous time step). That was from a talk I heard from a few years ago from so the details are a bit hazy. Nonetheless, the issues with shocks are HEAVILY studied.... for various military purposes.
If you aren't dealing with shocks and have a somewhat regular spatial domain, pseudo-spectral methods (i.e. using Fourier transforms) will give you good results. Just thinking about it heuristically, you should be able to get an idea for an adequate spectral band for your approximation if you can prove some type of gradient bound for analytic solutions to the problem. If there is no gradient bound (i.e. could mean a shock formation), this probably isn't the place to start.
If you are dealing with irregular domains then my guess is to look into FD methods using adaptive grids (i.e. densities would correlate to increasing gradients of numerical approximations). I assume you already know about implicit and explicit FD methods and various interpolations between the two... so I won't belabour that point.
I have no knowledge of using FE methods or of the level set methods that macklin01 mentioned in the setting of hyperbolic equations. Sorry.
You may also want to look into using fractional time steps. I.e. you take a half time-step using one numerical methodology, then you take the second half of the time step using another. Usually, the first half time step gives you a good approximation with less stability, the second half stabilizes the approximation from the first half. This is a very fast and loose explanation. Fractional time steps are well covered in literature (numerical recipes in C even briefly mentions them). Personally, I have found them very useful when using more exotic numerical methods that weren't so stable.
In regards of what to use to write the solver in. Personally, I always have used F90 or perhaps C to code up anything I wrote in this regard; but if you only plan on doing a few runs of the solver to get a few results, then matlab is a good fast alternative to F90. If you plan to run many simulations using your solver, then nothing will beat the speed of a well-coded solver written in F90.
As for visualization, time evolution of select level sets is the most obvious choice (like macklin01 said). It really depends on what you want to convey with your data though (as with any visualization). Thus, if you are trying to see just spatial representations of solutions, level sets are a good representation for 3D wave equations. You could also try color scales corresponding to your solution and looking at various slices of your domain; but this obviously requires some knowledge of where to look in your solution domain for certain phenomena. If you want to see shock formations, you could make 3D density plots based on the gradient value of your numerical app
I am currently a Ph.D. student in mathematics and have been teaching tutorials and courses since my second year of Uni. My feeling about using computers and various pieces of technology to teach either one of these topics is that they are counterproductive. I had two incredible teaches as an undergraduate in math and physics, and NEITHER used technology to make them effective teachers. What made them good was that they knew how to explain seemingly complex ideas in simple intuitive terms; and they actually helped students to visualize mathematics/physics in their head. They essentially taught students how to teach themselves new ideas.
The fact of the matter is, powerpoint style lectures have been shown to harm the attention span of students. It's like when you were in grade school and the teacher put on a movie for class; you zone out almost immediately. As for using computers to aid in 2D and 3D visualization, frankly I don't think this is a particularly good idea either. Students aren't always going to have a nice 3D movie given to them when they are faced with a new concept in math or physics. In higher level mathematics, naive 2D and 3D representations of concepts can be either misleading or incorrect; and making a student dependent on such simple visualizations cripples their ability to visualize more abstract concepts in effective ways. I believe in teaching students how to visualize things in their head; that is, how to sketch out pieces of a problem from something they see in the form of mathematics.
The point is, anyone can sit down and read a textbook... and if you are a teacher that essentially just regurgitates what is inside the textbook, you will never be a great teacher no matter how much technology you use. A great teacher or lecturer is the one who can show the student a path to original intuition. This will enable the student to draw their own visualizations, etc. when confronted with a new problem. This is precisely what made Richard Feynmann such an incredible teacher; he used intuition to make the most complex concepts intuitively clear to anyone... then he reconciled that intuition with mathematics. The mark by which I judge myself in understanding a subject is that I should be able to pull someone off the street and get at least the idea of what I'm talking about across to them in a few minutes. As I deal mostly in differential equations, I obviously can't do this just by writing equations; but it can be done intuitively by using simple explanations, heuristic examples and sketches.
I'm also a firm believer in the idea that students are meant to take notes in class. Writing information down in a note taking fashion forces the student to acknowledge the content being taught on some level. If nothing else, it reinforces what the lecture is saying. I know taking notes from a lectures that just dumps out information is difficult; but I believe them to be remarkably effective when paired with a teacher who conveys more in the way of intuition than statements of fact. The textbook is there to give the data... as a teacher you are meant to give the intuition to the student so they can sit down and understand that data. Forcing them to write down notes on their own will enable them to build a more solid connection with the intuition you are (hopefully) conveying.
In the end, learning by rote is the lowest form of understanding. Next comes the ability to do problems that you are already acquainted with. After this level comes the ability to solve slightly new problems by combining methods you used to solve previous problems. After this level, one gets into various levels of intuition which enable one to solve completely new problems. I try to teach my students in the hopes that they are seeking at least this level of understanding. I wish more teachers would spend more time working on their lectures along these lines as opposed to spending so much time and money to make their lectures nothing more than a powerpoint presentation.
I know this may come off
I can't believe that couples actually argue over such trivial crap. My wife and I have been nigh unseperable since before we were married (almost a year now). Yes, our marriage is still fairly new; but we both have previous experience with relationships that have lasted over the course of years. Since being married we've had to deal with several situations that I believe would strain the strongest of marriages. First, I moved to Australia a few years ago from the US; bringing my wife (also an American) to live with me here was no easy task. Issues and costs associated with this move alone were difficult for a low-income couple like us. We've had to deal with issues regarding families, money, jobs, housing, and even education (I'm finishing post-grad studies) since being married; and we've not had the altercations listed in this article or the slashdot posts.
Sharing a blog, emails, Tivo?! Who the hell really cares about this type of thing? My wife and I have separate emails just because it's confusing if we combine them. My wife knows all my passwords and I hers; so we can access each other's computers, accounts, whatever if we really need to. We help each other with our computers as well... she's an interface designer and I like playing around with hardware, kernels etc. It works out really nicely (we both are Arch users). We both have almost identical tastes in music, books and movies; this was one of the biggest reasons we hit it off so well in the beginning. Now I admit, I lucked out having a wife who actually likes the movie "Doom" and is a huge Tool fan; but I would have thought a common interest in these things would be had by many married couples. While we both like videogames, the only TV shows we really care to watch are Southpark and BSG. Again, I know I'm lucky; but then again, I didn't choose to marry someone with whom I really didn't have that much in common.
What I really think is the bigger issue here is porn. I, like every guy here, had a big porn collection before meeting my now wife. Being single, I was perfectly happy wacking off to porn whenever I felt like it; and I still don't see a problem with this. That being said, whenever I started dating a woman in the past; I would stop looking at porn on my own. I look at it like this, I would not be happy to walk in and see someone I'm dating masturbating to internet porn; so I don't do it. I have never been asked to stop looking at porn by anyone I've dated; it's just something I did on my own. In my view, masturbating to porn is the most effective way to make your partner feel utterly unattractive, and it will build resentment. I really feel that it's this building resentment that fuels arguments over trivialities; people don't want to take on the nasty issue, so they just berate each other over trivialities. There are many reasons why I really like my wife; but, one big one, is that are views on sexuality are the same. We both genuinely like sex as both a physical and mental act; and we don't deny each other's physical or mental desires. I believe this is healthy and unfortunately, somewhat rare given how twisted our societies' view on sex is. As for having sex with the same partner getting old; well, I don't really experience that. People's minds and bodies are infinitely fascinating; and so, I haven't ever felt boredom with a sexual partner even if we've been together for years.